Six Phase Power Transmission System

IN THE NAME OF ALLAH, THE MOST BENEFICENT THE MOST MERCIFULREAD: In the name of your LORD Who created, created man from a clot Read: and your lord is most Bounteous Who taught by the pen Taught man that which he did not know. Taught man that which he did not know. Surah Al-Alaq (Al-Quran) Verse # (1-4) Chapter # 30 Simulation and Analysis of Six-phase Power Transmission System Session 2008-2012 Group Members Safdar Rasool Muhammad Kashif Nadeem Muhammad Awais Rafique Aamar Iqbal 2008-RCET-ELECT-02 2008-RCET-ELECT-06 2008-RCET-ELECT-16 2008-RCET-ELECT-22 Project Supervisor Engr. Rehan Arif Department of Electrical Engineering Rachna College of Engineering and Technology, Gujranwala (A Constituent College of University of Engineering & Technology, Lahore) i Simulation and Analysis of Six-phase Power Transmission System Submitted to the faculty of the Electrical Engineering Department of the University of Engineering and Technology Lahore in partial fulfillment of the requirements for the Degree of Bachelor of Science in Electrical Engineering Approval on _________________ External Examiner External Examiner External Examiner Internal Examiner Department of Electrical Engineering Rachna College of Engineering and Technology, Gujranwala (A Constituent College of University of Engineering & Technology, Lahore) ii Kashif Nadeem M. Safdar Rasool M. In addition this work has not been submitted to obtain another degree or professional qualification. except where explicitly stated otherwise.Declaration We declare that the work obtained in this report is our own. Awais Rafique Aamar Iqbal 2008-RCET-ELECT-02 2008-RCET-ELECT-06 2008-RCET-ELECT-16 2008-RCET-ELECT-22 _______________________ _______________________ _______________________ _______________________ iii . loving parents. keen interest. encouragement. His sweet behavior. hearty sentiments and thanks to our project advisor Engr.Acknowledgment All glory to Almighty Allah. co-operative sisters and brothers and opportunity to make this humble contribution and all praises to.H) Whose blessings and exaltations flourished our thoughts and thrived our ambition to have cherished fruit of our modest effort in form of this write-up. the creator of this universe. knowledge delivering. The Gracious and compassionate whose bounteous blessings gave us potential thoughts.U. We would not have been able to complete our project without his supervision. We express our most sincere gratitude. personal involvement and criticism for the betterment were all the real source of courage. helping friends. respect and ‘Darood-O-Salam’ are due to His Holy Prophet(P. inspiration and strength during the completion of this project. Rehan Arif for his excellent supervision. talented teachers. iv .B. Dedicated to… GREATEST REFORMER HAZRAT MUHAMMAD (PBUH) OUR PARENTS WHO‟S PRAYERS ARE FOR US OUR TEACHERS WHO ENCOURAGED US AT EVERY POINT OUR BROTHERS AND SISTERS WHO’S INNOCENT SMILES ARE FUEL FOR OUR LIFE. v . .........4 2...............................................................4 2........................................................................3 Introduction .................................1 2..........1 1.... iv Dedicated to… ................................3................... 13 Higher Power Transfer Capability ................................................... 12 Advantages of Six Phase Power Transmission ............................ 2 Objectives and Scope ....................................................................... xv Abstract ......................................... 1 1...........................................................................................................................................3 2............................5 2....................................................................................................................................................................................................................... xiv List of Symbols and Acronyms ......................................... x List of Table ....................................... 9 Phase-to-Phase Voltage ................................. 8 Phasor Relationship in Three-Phase System ............................................................................................ xvii Chapter 1 Introduction .................................................................................................................... 7 Phasor relationships ...................................................5 Research Background .................................................................... 10 Phase-to-Group Voltage .....................................1 ....3..... v List of Figures...... 6 2.....Table of Contents Declaration..............2 1...........................3................3.. 6 Voltages in Six Phase System .............................................................. iii Acknowledgment.................... 12 2.............................. 5 Chapter 2 Six-phase Power ............................1 2..................................................... 13 vi 2.........................................................4 1.......................3............................................. 8 Phasor Relationship in Six-Phase System . 1 Literature Assessments ............................................................................................................5 Power in Six Phase System............................................................................................................................................... 11 Phase-to-Cross phase Voltage ................5.2 2..... 4 Thesis structure .................................2 2........................................................ ...................................................................... 23 ∆-Y Connection ................................................6 2..................................................................4. 17 3...................................5....2 3............................... 21 Y-Y Connection .......3................................................................2 ...........6 2.................3 3..............2 2.............................1.............................................................................................................5 3........ 19 3........................................ 15 Lower Corona and Field Effects .................4 3.3 3................1 Production of Six phase .................... 27 Diametrical ........................3 Power Transformer ......................5.............4.....................................4 2........ 16 Chapter 3 Production of Six Phase Power and System components ..............................................................1 3............................... 15 Lightning Performance .... 16 Summary ........................3 2...................................................................................................... 15 Feasibility ....... 33 Surge Impedance Loading...............5. 19 Three-Phase Transformer Connections ....7 2.2 3..............................5.........................3..5...........................................................2 3......... 28 Double-Delta ... 31 3................ 18 Direct Six-phase Generation ........................... 14 Lower Insulation Requirement ................................2 3............................................................4......................1 3.........4 3... 26 ∆-Y and ∆-Inverted Y .................................................................................................3.................. 33 vii 3.......5 2....................... 14 Smaller Structure ..4 Six-Phase Transformer Connections........................................7 Increased Utilization of Right-of-Way..3....................4.... 18 Three-phase to Six-phase conversion .......................................................................................................................4.......................................................... 24 ∆-∆ Connection ......... 31 Surge Impedance ..........1 3...............5 Power Transmission Line ................................ 29 Double-Wye .......... 22 Y-∆ Connection ...............5.....1................... 25 3.................................... 25 Y-Y and Y-Inverted Y ..............................................................................................................................5..............................................................1 3...........2.................5.......................... 15 Better Stability Margin ......... ...1.. 56 Chapter 5 Electromagnetic Field Gradients ..... 58 5.. 64 Results and Conclusion ....................... 55 Summary ..1.1 4.................................. 36 4..................2 Application to Transmission Lines ............................................................5........ 65 5.......2...............................................4 3................................................................................2 4....................... 62 Magnetic Field of Six-phase Line with same load .....6 Line Loadability ....... 34 Stability Performance .........3 3...................3 5............................. 46 4................... 63 Magnetic Field of Six-phase Line with Increased load ......2 conductor ......................1...................................5 The Role of Simulation in Design .................5........................................................................1 Magnetic Field Basics.....................4 5....... 38 Modeling of Three-phase double circuit line on Simulink ...3................................................... 59 Computer Program for calculation of Magnetic Fields ..............................................................2..................... 62 Magnetic Field of Three-Phase Double Circuit Line.1 5.....................................5.4 viii .........................................2 4............ 44 4...............................................................................................................1 5.............................1................................. 35 Chapter 4 Modeling of six-phase Transmission System in MATLAB® .......................5..............................................3 4...................... 57 5.............................. current carrying 5.......... 35 Summary .....3 5............6 4..........................1 Transformation block for wye-wye wye-inverted-wye ..... 59 Magnetic field strength for Six-phase Line .......2 5. 50 Voltage Drop Comparison .............................................................4 4.2..........................................7 Delta-wye Delta Inverted wye configuration ... 57 Basic Concepts: ................ 40 Modelling of Six-phase Transmission System ......... 58 Application of Ampere’s Law to infinitely long........................................2......................... 36 SimPowerSystems ............................... 36 Overview of SimPowerSystems Libraries .................................................................. ........................... 82 Chapter 6 System Modifications and Cost Analysis...........................................3 Cost Analysis ................................................. 91 Project Limitations and Future Recommendations ..................................... 66 5... 80 Results .............................................................................................2 5.....1...................................................................................... 87 6..............8 6.........................2 6.............................. 78 Corona loss Calculations ......... 86 Insulation Requirements.... 86 Tower Structures ...................4 5...................1 7.................................................................................. 82 5..1 System Modifications ................. 67 Computer Program for calculation of Electric Fields ...1................................ 85 Protection.............5................ 76 Corona .....................................4 Basic Equations ............................................... 84 Six Phase Positioning ................4...... 79 Corona Precautions for Compact Lines .......... 98 ix ................................. 95 Appendices ......................................................................................... 93 References ........................................... 86 Right of Ways ........................5 6......................................... 87 Summary ....... 90 Chapter 7 Conclusions and Future Recommendations ................................................. 83 6...3.............................................................3 6.1.....................3 Analysis of transmission line conductor surface voltage gradients computations ............7 6..................2 5..............5 Summary ....................................1.......................................................3 5.......................................................................3..................................2 6...............................................................4 6.......................................1..............................................................................1.... 85 Transmission line Modifications ........................................................1............................................6 6.....4................................................................................................... 84 Six-Phase Conversion Transformers ...1 5..................................................1 6.............2 Results and Conclusions ........................................................................4....................................................................... 91 7......1................................................. 84 Six-phase Bays ........... .28 x ....26 Figure 3.……22 Figure 3.8 Figure 2......................17 Figure 3.......15: Schematic diagram of ∆-Y and ∆-Inverted Y connected three-to-sixphase conversion transformer…………………………………………………….…...…………9 Figure 2.....……….....25 Figure 3..11 Figure 2..List of Figures Chapter 2 Figure 2...20 Figure 3..7: Determining power density…………………………………………..3 Phasor diagram of three phase system………………………......27 Figure 3..1: Machine Power Vs No.13: Schematic diagram of Y-Y and Y-Inverted Y connected three-to-sixphase conversion transformer.......4:Y-Y connected three-phase transformer…………………………......23 Figure 3.………18 Figure 3..7: Schematic diagram of Y-∆ connected three-phase transformer……..27 Figure 3...3: 20 MVA three-phase transformers………………………..5: Schematic diagram of Y-Y connected three-phase transformer….5: Potential between phase A and phase C………………………………11 Figure 2...9: Schematic diagram of ∆-Y connected three-phase transformer………24 Figure 3.14: ∆-Y and ∆-Inverted Y connected three-to-six-phase conversion Transformer………………………………………………………………………..7 Figure 2.8: ∆-Y connected three-phase transformer………………………………24 Figure 3..10: ∆-∆ connected three-phase transformer…………………………….28 Figure 3.……..14 Chapter 3 Figure 3. of Phases……………………………………...........2: DGC Triangle representing relationship between Vphase and Vline….…......16: Diametrical connected three-to-six-phase conversion transformer……………………………………………………………………….6: Potential between phase A and phase…………………………………12 Figure 2.1:Phasor Diagram of Six-Phase System…………………………………...........4: Potential between phase A and phase B………………………..2: Six-Phase double wye Synchronous Generator……………….22 Figure 3.12: Y-Y and Y-Inverted Y connected three-to-six-phase conversion Transformer……………………………………………………………………….....11: Schematic diagram of ∆-∆ connected three-phase transformer…..6: Y-∆ connected three-phase transformer………………………………23 Figure 3..25 Figure 3. Figure 3.17: Schematic diagram of Diametrical connected three-to-six-phase conversion transformer……………………………………………………………29 Figure 3.18: Double-Delta connected three-to-six-phase conversion transformer………………………………………………………………………...29 Figure 3.19: Schematic diagram of Double-Delta connected three-to-six-phase conversion transformer……………………………………………………………30 Figure 3.20: Double-Wye connected three-to-six-phase conversion transformer………………………………………………………………………...30 Figure 3.21: Schematic diagram of Double-Wye connected three-to-six-phase conversion transformer……………………………………………………………31 Figure 3.22: Lossless line terminated by its surge impedance.................................33 Figure 3.23: Surge impedance loading characteristic curve………………………34 Chapter 4 Figure 4.1: Nonlinear Simulink Blocks for SimPowerSystems Models…………..39 Figure 4.2: Simulink Library Browser…………………………………………….39 Figure 4.3: Display block for numeric display of input values……………………40 Figure 4.4: Block diagram of Three phase transformer …………………………..40 Figure 4.5: Block Diagram and Connection Diagram of Three Phase T/F……..…41 Figure 4.6: Transmission Line…………………………………………………….41 Figure 4.7: Waveform of Phase Voltages………………………………………....42 Figure 4.8: Waveform of Line Voltages………………………………………..…42 Figure 4.9: Hierarchy of Measurement blocks for Phase Voltages…………….….43 Figure 4.10: Hierarchy of Measurement blocks for Line Voltages………………..43 Figure 4.11: Complete model of Three Phase double circuit Transmission System.......................................................................................................................44 Figure 4.12: Three-Phase RLC load………………………………………………44 Figure 4.13: Y-Y Y-Inverted Y Configuration of Transformers………………….46 Figure 4.14: Waveform of Phase voltages………………………………………...47 Figure 4.15: Waveform of Line Voltages…………………………………………47 Figure 4.16: Source Voltages………..…………………………………………….48 Figure 4.17: Voltages across Load……………………………………………...…49 Figure 4.18: Complete System for Six Phase Transmission Using Y-Y, Y-Inverted Y Transformer configuration……………………………………………………...49 xi Figure 4.19: Hierarchy of Delta-Wye Delta-Inverted Wye Transformation block………………………………………………………………………………50 Figure 4.20: Waveform of Phase Voltages………………………………………..51 Figure 4.21: Waveform of Line Voltages………………………………………....51 Figure 4.22: Three-Phase RLC Load……………………………………………...52 Figure 4.23: Six Phase Transmission System using Delta-Wye Delta-Inverted Wye Configuration of Transformer……………………………………………………..53 Figure 4.24: Diametrical Configurations……………………………………….…53 Figure 4.25: Block Diagrams………………………..……………………….........54 Figure 4.26: The connection diagram of Diametrical conversion transformer……54 Figure 4.27: Input parameter of transmission line………………………………...55 Chapter 5 Figure 5.1: The BiotSavart Law…………………………………………………...58 Figure 5.2: Magnetic field of aconductor along Z-axis carrying current I………...59 Figure 5.3: Magnetic Field of a single conductor…………………………………59 Figure 5.4: Magnetic field of a multi-conductor line……………………………...60 Figure 5.5: Relation between the lengths and Tower Geometry……………..........61 Figure 5.6: Magnetic Field Profile of Three-phase Double Circuit Transmission Line…………………………………………………………………..……………62 Figure 5.7: Magnetic field of three-phase double circuit transmission line….........63 Figure 5.8: Magnetic field Profile of Six-phase line with same Load…………….64 Figure 5.9: Magnetic Field Profile of Six-Phase with increased load……………..65 Figure 5.10: Plot of Magnetic field of six-phase line……………………………...65 Figure 5.11: Vector addition of field due to two charges…………………….........68 Figure 5.12: Potential difference between two points a and b…………………….69 Figure 5.13:Linear path in nonunform electric field……………………………....69 Figure 5.14: Transmission line of n-conductors…………………………………..71 Figure 5.15: Electric fireld produced by source and image conductor……………73 Figure 5.16 n-conductor system………….………………………………………..75 Figure 5.17: Electric field profiles………………………………………………...77 Figure 5.18: Plot of Electric Field versus Distance for Three-Phase……………...77 Figure 5.19: Plot of Electric Field versus Distance for Six-Phase………………...78 xii Chapter 6 Figure 6.1: Plot of Total Line Costs for Six-phase and three-phase double circuit lines. ……………………………………………………………………………....89 xiii 1 Transformer configuration ……………………………….79 Table 5.5: Results for corona loss…………………………………………………80 Chapter 6 Table 6.56 Chapter 5 Table 5.2: Input Data for Six-phase line with same load………………….………63 Table 5.1 Goudy-Oakdale lightning performance flashovers per year for 2...………………………………………………….2: Cost of the equipment for upgrading of three-phase double circuit line.3 Voltage drops across the length of transmission line for Six phase with same load as three-phase…………………………………………………………..89 xiv .2 Voltage Drop across the length of transmission lines for three-phase and six-phase with 73% extra load.62 Table 5.1: Cost for the Equipment to be installed in Six-Phase line………………88 Table 6.4: Line Configuration and Conductor Data………………………………..1: Input data for three-phase Double Circuit Transmission Line………..……………46 Table 4.List of Tables Chapter 2 Table 2.4 km of line ……………………………………………………………………………….56 Table 4.16 Chapter 4 Table 4..3: Input Data for Six-phase line with 73% increase in load………………64 Table 5... New York State Energy Research and Development Authority TPDC. Hz Machine rating in.List of Symbols and Acronyms αδθπωaAC CDC EfGHAngular acceleration.A MATLAB™ Power System Simulation Package Nn– Number (of phases/phase conductors. turns. etc.Allegheny Power Services Corporation DOE . degree Angular displacement.Generator step-up HPO . MVA Inertia constant or Height.Extra-high voltage GSU . joule-sec/radian MATLAB .Matrix laboratory software MATPOWER . m APS . radians/second Transformer turn ratio or 1 ∠ 120° in polar number Asynchronous current Capacitance.New York State Electric and Gas Corporation NYSERDA . radians/second² Angle difference between the voltages. μF Direct current Excitation voltage Frequency.Three Phase double Circuit PReal power xv .High phase order HVDC-High-voltage DC ILMCurrent Inductance. radians 3.National Transmission and Dispatch Company WAPDA-Water And Power Development Authority NYSEG .1416 radians or 180° Angular velocity. mH Angular momentum.Department of Energy EHV .) or Neutral Speed NTDC. m Positive-sequence impedance. Ω System reactance. Nm Voltage Phase-to-neutral voltage Phase-to-phase voltage UHV . Ω Admittance. Ω Leakage Reactance as seen from winding 1. mho Y-Y .Power System Computer Aided Design/ Electromagnetic Transient for Direct Current PTI .Phase-to-phase voltage at secondary side VPP .Wye-Wye connection of the transformer winding Y-Δ .Delta-Delta connection of the transformer winding zZc Impedance.Phase-to-neutral voltage at primary side VPS .Surge Impedance Loading Ta VVP VL Torque.Pa Pe Pm - Accelerating power Electrical output of machine Mechanical power input of machine PSCAD/EMTDC.Wye-Delta connection of the transformer winding Δ-Y .Ultra-high voltages VLP .Power Technologies Incorporated SKVA-Three-phase apparent power. Ω Positive-sequence surge impedance of the line.Phase-to-phase voltage at primary side VLS .Phase-to-neutral voltage at secondary side Wxxe xs XL yWide. kVA SIL . Ω Generator synchronous reactance. Ω xvi .Delta-Wye connection of the transformer winding Δ-Δ . These studies have been performed in sufficient detail to determine how the six-phase conversion will affect steady state operation and system stability. which is undergoing rapid industrialization. One of the main advantages of six-phase transmission is that a six-phase line can carry up to 73% more electric power than a three-phase double-circuit line on the same right-of-way of transmission line. In this project. load flow results shown the voltage levels and voltage phasors are also discussed. EHV systems have been growing rapidly and widely throughout the world because of their efficiency and economy but EHV systems might have adverse impacts on environment like corona loss. effects of electric and magnetic fields are also included in this project. So. it has been shown that the Test Systems with six-phase single-circuit transmission has a better stability limits compare to the three-phase double-circuit transmission in the sense of power transfer capability and voltage drops.Abstract Electricity is considered as the dynamo for a country. this conversion will have impacts on the power system operations. Constrains on the availability of land and planning permission for overhead transmission lines have renewed interest in techniques to increase the power carrying capacity of existing right-of-ways (ROW). audible noise and formation of noise. However. respond to the worries related to electromagnetic fields. radio interference. Six-phase transmission appears to be the most capable solution to the need to increase the capability of existing transmission lines and at the same time. investigation is made in time domain considering conversion of three-phase double-circuit to six-phase single-circuit transmission system by using SimPowerSystems in MATLAB/Simulink® program. In the end justifications are made for the extra cost of conversion and inversion units for generation of six-phase power in transmission systems. xvii . Besides. From the simulation results. D. The increased interest in HPO electric power transmission over past thirty years can be traced on a CIGRE paper published by L.S. With the increase of energy demand as rapid growth of World’s economy has caused an increased on the demand of electricity supply and load currents of transmission lines. R. Among the HPO techniques. Its need for industrial use is increasing day by day. 6-Φ transmission is proved to be most reliable for increasing the capability of existing transmission lines and at the same time it deals with electromagnetic fields as well.e. increase in power transmission capability has been accomplished by increasing system voltages. However. That requires new generators and transmission systems to be installed. engineers are looking for some alternative i. Since then. Due to the high costs involved in the installation of new transmission lines.C. Barthold and H. One of the main advantages of 6-Φ transmission is that a 6-Φ line can carry up to 73% more electric power than a 3-Φ double-circuit line on the same transmission 1 .1 Research Background Electric power has become a basic need of humanity. the concept of HPO transmission has become vast and it is being described in several papers and reports. In the past. In high phase order. to enhance the power transfer capability in the existing system. the enhanced power system capability with the increase in 73% load was discussed by A.Chapter 1 Introduction Chapter 1 Introduction 1. Kelkar [1].B. Barnes. increasing of transmission operating voltage will produce strong electric and magnetic fields at ground level with possible biological aspect and environmental effects which necessitate large Right-of-Way (ROW). Pandya. Mishra.2 Literature Assessments A lot of work is done on high phase order as during 1981-83 Dr P. the current research results to have a better picture and clearer understanding of the 6-Φ power transmission system. six phase transformers These analyses will be performed on various test systems which include IEEE Test Systems in detail using simulation program like MATLAB. reactive power limit in 6 phase is increased at each point of receiving end voltage [3]. study of the analysis of six phases is accomplished during normal operating conditions for electric power system considering 3-Φ-to-6-Φ conversions of selected transmission lines in an electric energy system. as they are applicable to a particular simple system. Magnetic fields 3. maximum power at the receiving end will progressively enhance maintain the voltage stability at various power factors of load.. investigated and found different methods of 6Φ systems. Right of ways 4. It was found that conversion of an existing 3-ɸ double circuit to 6-ɸ single transmission line results in line inductance increment and capacitance decrement.S. 1.K. For this reason. 1.A.Chapter 1 Introduction [2]. Power transmission capacity 2. S. for the length about higher then 160KM. Cost effectiveness 5. and calculations of inductance and capacitance values for 6Φ lines. Six phase transmission is conceived as a technique to increase the power transfer capability of existing ROW space. During 1993-94 Mr. how much 6-Φ conversion will affect steady state operation. In this research. and system stability. 6-ɸ single line conversion.Subramanyam. 2 .Chandraserkharanetal. it is found that voltage stability as a recent challenging subject was analyzed. Following analysis will be performed to know.Venkatetal. fault current duties. Also. mathematical modeling of 3Φ/6Φ transformers. carried transient stability analysis of a 6Φ line using the standard Byrd & Pichard equation which yields closed form expressions and lacks the generality. However the minimum line length at which power transfer capability is limited by voltage stability concern is dramatically decreased in 6-ɸ single line compared to 3-ɸ double circuit due to conversion transformers reactance effect. Moreover.S. Mr. Increment in transmission distance and transmission capacity For the purpose of transmitting power over very long distances. analysis of system characteristics and system protection.Chapter 1 Introduction The incentives for increasing transmission voltages have been: 1. The main disadvantage of the DC is the production of harmonics which requires filtering. showed that the HPO transmission should be considered as a viable alternative to the conventional 3 . W. the first modern High-Voltage DC (HVDC) transmission line was put into operation in Sweden between Vastervik and the island of Gotland in the Baltic Sea. it may be economical to convert the EHV AC to EHV DC. HVDC lines have no reactance and are capable of transferring more power for the same conductor size than AC lines. This is based on the fact that. using converters we first convert AC to DC and invert it back to AC at the other end. much has been added to the available knowledge base on HPO transmission primarily in the areas of feasibility considerations. Reduction in ROW 2. and a large amount of reactive power compensation required at both ends of the line. Allegheny Power Services Corporation (APS) in cooperation with West Virginia University began seriously investigating the details of an HPO designed in1976. Reduction in line losses 4. The work had focused the industry on the practical aspect of concepts that were first explained by Fostesque [5] in 1918 and E. Since this corner stone work. Their studies. funded partly by the National Science Foundation. Lower capital and operating costs of transmission. The DC transmission tie line acts as an asynchronous link between the two rigid systems eliminating the instability problem inherent in the AC links. In the late 1970s. Clark[6] in 1943. C. Smaller line-voltage drops 3. One variable which relates to that efficiency is the number of phases. Guyker [7] extended the transmission concept by describing fault analysis methodologies and symmetrical component theory. 5. In 1954. They also assessed the feasibility of upgrading an existing 138kV line to 6-Φ to increase the power transmission capability by 73% while reducing conductor field gradients and improving system stability which potentially could obtain public acceptance the nominal voltage of the line would remain unchanged. DC transmission is especially advantageous when two remotely located large systems are to be connected. the EHV DC has lower losses in transmission line and also has no skin effect [4]. in addition to enhanced power transmission capability. 4 . thus eliminating the incentive to pursue increased power transfer capabilities. provides low voltage gradients. increased utilization of ROW. lower insulation8 requirements and better voltage regulation. According to new idea the feasibility of 6-Φ transmission system is represented in terms of insulation performance. Simulation has been performed on these two transmission lines.4 Objectives and Scope The objective of this project is to provide a solution for the limited Power Transmission Capability of existing transmission lines and to eliminate the legal and environmental constraints involved in the construction of new transmission lines in the form of Electric and Magnetic field Gradients and Right of Ways respectively. corona and field effects. APS covered the way for future research. Further. Load projections for their service area were reduced. 1. They completed detailed analysis of HPO designs and protection philosophies. lower corona effects. The project was abandoned. Some of the advantages of using the 6Φ transmission system are increased transmission capability. Experiences with the use of the PSCAD/MATLAB software have been positive and have enhanced the quality of research and teaching. This study has given verification to available methods for the calculation of electric and magnetic fields.Chapter 1 Introduction 3-Φ transmission system. it has been shown that the 6-Φ transmission system can provide the same power transfer capability with lower ROW or can transfer 73% more power for the same ROW as compared to the 3-Φ double-circuit system. smaller tower structures reduce the right of way requirements. Comparative studies for 3-Φ double-circuit and 6-Φ single-circuit transmission lines have been implemented to get better one out of the two for future projects. Six-phase transmission. In this project the models of 3-Φ double-circuit transmission and 6-Φ single-circuit transmission models by has been developed using MATALB program. but stopped short of actually demonstrating the technologies on an operating line. the simulation based approaches proved to be very effective. however through their initiative. radio noise and audio noise from the 6-Φ overhead lines. and load flow and system stability. Besides. Chapter 5 states that size of insulator required in six phase transmission towers will be less as compared to the three-phase double circuit and size of tower will also be compact as ground clearances and mid span clearances will be reduced. radio interference. Here we have established the definitions for system Voltages. Power and Phasor Relationships. we establish the methods of production of Six phase power and components used Six phase power Transmission system that include six phase generator. In this chapter 6. In this chapter 2 we have discussed the six-phase power system in detail. 5 . Also it is concluded that electric field is less for 6-ɸ than 3-ɸ. and analysis of simulation of 3-ɸ to 6-ɸ conversion of selected transmission line in electric power system. Load flow analysis and power transfer capability comparisons are also performed. modeling. Eventually. TV interference and formation of ozone due to corona will also reduce as electric field strengths are diminished.5 Thesis structure This thesis is primarily concerned with the understanding. six phase transformer and six phase transmission line. corona loss. All the work is this research is presented in chapter 4 . we first discuss the modifications required in conversion of a three-phase double circuit transmission line to a six-phase line and discussing the savings/expenses in terms of cost in all the equipment. 5 and 6th. In chapter 4 modeling and comparison of three-phase double circuit and six-phase single circuit are performed in Simulink /MATLAB®. In chapter 3.Chapter 1 Introduction 1. Later a cost analysis is performed in which a 500kV six-phase line is compared for relative economics with a 500 kV three-phase double circuit design. For a Six-phase this becomes six equal magnitude voltage vectors spaced 60o between adjacent phases and so on.1 Introduction In recent years. HPO is defined by number of phases of having equal magnitude of voltage but equally spaced in time. the key to the benefits of HPO transmission system lie in the Line and Phase voltage relationships. increase in power transmission capability has been accomplished by increasing system voltages. In the past. As discussed earlier.Chapter 2 Six-phase Power Chapter 2 Six-phase Power 2. Here we have established the definitions for system Voltages. In this chapter we have discussed the six-phase power system in detail. 6 . In consideration of the fundamental limits on power transfer capability in a restricted ROW led to the concept of increasing the number of phases in a transmission line system circuit also known as Multiphase system or High Phase Order (HPO) Transmission system. [8] However. Availability of power at generation stations has caused an increase in load currents of transmission lines to supply the growing load. rapid growth of World’s economy has caused an increase on the demand of electricity supply. this means three equal magnitude voltage vectors spaced 120o from each other. [9] For three phase system. Six phases have attained more importance than other HPO systems because of its feasibility in application on existing system that is a Three Phase Double Circuit (TPDC) Transmission Line can be converted into a six phase line without making extraordinary modifications. increasing of transmission operating voltage will produce strong electric and magnetic field at ground level with possible biological aspect and environmental effects which necessitate large Rightof-Way (ROW). Power and Phasor Relationships. So with this commitment voltage of group I and II belongs to the Vphase and Vline respectively. VCF Here we define Vline as the voltage between the adjacent phases and Vphase as the voltage between a phase and ground. VDG. In the groups I and II the voltages are spaced 60o. VDE. VCD. VEA. Figure 2. in the group III and IV the voltages are spaced 120o and 180o respectively: (i) Group I (phase-to-ground voltage): VAG. Within each group.1. VCG. Voltage between adjacent phases. the voltage system can be classified into four discrete voltages. having such a Voltage on all the phases which is equal in magnitude but spaced at an electrical angle of 60o from each other. that are Phase-to-ground Voltage. and Voltage between opposite phases.1. VAD. there are six phases.2 Voltages in Six Phase System In six phase power system. VFG (ii) Group II (between adjacent phases) VAB. VBC. Voltage between phases separated by one intermediate phase. The equation of Vline and Vphase can be derived by determining the resultant of DGC triangle in Fig 2. VEG. VEF. VFA (iii) Group III (between phases separated by one intermediate phase) VAC. Phasor diagram of phase-to-phase and phase to ground for a six phase system is shown in Figure 2.1) .1: Phasor Diagram of Six-Phase System From Fig 2.2 [10]: VCD = 2 x VCG’ = 2 x VCG Cos θ 7 (2. VCE. VDF. VBG. VBD.Chapter 2 Six-phase Power 2. VFB (iv) Group IV (between opposite phases). VBE. all the voltages have identical magnitudes. Chapter 2 Six-phase Power Angle θ for adjacent phase-to-phase is 60o. The phase-to-phase voltage is 3 of the phase-to-neutral voltage.3 Phasor relationships As we have to carry out our discussion for three phase double circuit transmission line side by side with the Six-phase transmission line.3.6) VBN = VBN -120° VCN = VCN 120° Assuming the VAN = VBN = VCN = VP (i.1 Phasor Relationship in Three-Phase System A typical balanced three-phase system has 120o electrical degrees between each phase as shown in Figure 2. their phasor diagrams and relationships are discussed in detail 2.3.VBN -120° 8 (2. Generally phase-to-neutral voltages.7) .2: DGC Triangle representing relationship between Vphase and Vline Hence. From Figure 2. Vphase = Vline (adjacent) in the following topic. if VAN is assumed as reference can be described as: VAN = VAN 0° (2.4) (2. VAB = VAN 0°. Voltage Magnitudes of all the phsors are same).3) The rest of voltages. 2. so we first establish a phasor relationship for three phase system and then extend our discussion to six-phase power system. e.5) (2. we can obtain the relation of phaseto-phase Voltage and phase-to-neutral voltage.3. (2. it can simplified that Vline (adjacent) = VCD = 2 x Vphase Cos60o (2.2) Figure 2. with 120° between phases has a phase-to-phase voltage equal to kV. First group has 60°. So in this way three different groups of voltage related with other voltages arise as way have already discussed in above article. then the phase-to-neutral voltage is 76.9) A three-phase system.(-0.2 Phasor Relationship in Six-Phase System A balanced six-phase system has 60o electrical degrees between each phase as shown in Fig 2. VAN is assumed as reference. VFB. c) Phase-to-cross phase voltage.5 .3. The voltage relationship for the phases in a six-phase system represented by above three groups refers to the phase shift between all six lines. VL-Group : VAB. VCE. 3 phase-to-neutral voltage and always leading phase-to-neutral voltage by 30°.3: Phasor Diagram of three-phase system The same correlation is applies for phase-to-phase voltage VBC and VCA. VL b) Phase-to-group voltage. VBD. The groups are as follows: a) Phase-to-phase voltage. VEA. VDE. : VAC. Generally phase-to-neutral voltage. VBE. the relationship between phase-to-phase voltage and phase-to-neutral voltage is given as follow: VL = 3 VP 30° (2.2 2. If the phase-to-phase voltage is 132 kV. VCF. VCD. second group has 120° and last group has 180° phase shift between phases. VFA. VBC.5 + j0.8) Figure 2.866)) = VP (1. VDF.866) = 3 VP 30° (2. In general.1 -120°) = VP (1+j0 . VEF.j0. VL-Cross phase : VAD.Chapter 2 Six-phase Power = VP (1 0°.1. 9 . j0. VEF and VFA. In general.866)) = VP (0.3. Assuming the VAN = VBN = VCN = VDN = VEN = VFN =VP.4 shows the potential between phase A and phase B.3 Phase-to-Phase Voltage We have already obtained a relationship between phase-to-phase voltage and phaseto-neutral voltage for a six-phase system mathematically. VBC. the magnitude of phase-to-phase voltage is equal to the magnitude of the phase-to-neutral voltage and phase-to-phase voltage always leading the phase-to-neutral voltage by 60°.5 .VBN 0°. the relationship between phase-to-phase voltage and phase-to-neutral voltage is given as follow: VL = VP 60° (2. Phase-to-phase voltage is a potential between adjacent phases where their phase difference is 60°.10) The same correlation is applied for phase-to-phase voltages VAB.11) 10 .866) = VP 60° (2. VCD.5 + j0. VAB = VAN = VP (1 0°.Chapter 2 Six-phase Power VAN = VAN VBN = VBN VCN = VCN VDN = VDN VEN = VEN VFN = VFN 0° -60° -120° -180° -240° -300° 2.(0. Fig 2.1 -60° -60°) = VP (1+j0 . VDE. Now we obtain the same using an alternate method. For a six-phase system. Chapter 2 Six-phase Power Figure 2. Fig 2. VBD.neutral voltage by 30°.866)) = VP (1.VCN ∠ -120° 0°. VCE.(-0.5: Potential between phase A and phase C 11 . the relationship between phase-togroup voltage and phase-to-neutral voltage is given as follow: VL-Group = 3 VP 30° (2.3.j0.5 + j0.4: Potential between phase A and phase B 2. VEA and VFB.866) = 3 VP 30° (2.4 Phase-to-Group Voltage Phase-to-group voltage is a potential between phases where the phase difference is 120°.13) Figure 2. VAC = VAN = VP (1 0°. The magnitude of phase-to-group voltage is 3 times the magnitude of the phase-to-neutral voltage and phase-to-phase voltage always leading the phase-to. VDF.5 shows the potential between phase A and phase C.1 -120°) = VP (1+j0 . In general.5 .12) The same correlation is applies for phase-to-phase voltages VAC. 0 + j0)) = VP (2) = 2VP 0° (2. the relationship between phase-to-crossphase voltage and phase-to-neutral voltage is given as follow: VL-Crossphase = 2VP 0° (2. keeping Vphase to neutral (3 phase) there is no increase in power. The magnitude of phase-to-cross phase voltage is two times the magnitude of the phaseto-neutral voltage.6 shows the potential between phase A and phase D. Pthree-phase-double-circuit = 2 (3 Vphase-to-neutral Iline) = 6 Vphase to neutral (3 phase) Iline Whereas power in Six-phase line can be calculated as: PSix-phase = 6 Vphase to neutral (6 phase) Iline equal to Vphase to neutral (6 phase). (2.VDN 0°. but the 12 .3. In general.(-1.17) If a three-phase double circuit line is upgraded to a six phase line.15) Figure 2.16) (2. Fig 2.6: Potential between phase A and phase 2. VBE and VCF.1 -180° -180°) = VP (1+j0 .4 Power in Six Phase System Assuming unity power factor power in a three phase double circuit transmission line can be calculated using following formula.5 Phase-to-Cross phase Voltage Phase-to-crossphase voltage is a potential between phases where the phase difference is 180°.Chapter 2 Six-phase Power 2. VAD = VAN = VP (1 0°.14) The same correlation is applies for phase-to-phase voltages VAD. the Right of Way (ROW) requirement is reduced due to the reduction in electric and magnetic field gradients. for the same phase-to-phase voltage as in the threephase system. consuming the same ROW and having same electric and magnetic field strengths. Therefore.16 can be written for six phase power as: PSix-phase = 6 Vphase to neutral (6 phase) Iline = 6( 3 Vphase to neutral (3 phase) ) Iline = = 3 (6Vphase to neutral (3 phase) Iline ) 3 Pthree-phase-double-circuit 3 (or 1. from equation 2. 2. if a Vphase to neutral (6 phase) is increased to Vline-to-line (3 phase). This 13 . when a three-phase double-circuit line is converted to six-phase line. This also results in smaller supporting structures. Vphase to neutral (6 phase) = Vline to line = 3 Vphase to neutral (3 phase) So.73) times higher than Vphase to neautral (3 phase).5 Advantages of Six Phase Power Transmission With the growing concern over the environmental effects of power system. it can be observed that. = 1.Chapter 2 Six-phase Power increase is in power density. That is. sixphase transmission offers several advantages over conventional three-phase doublecircuit networks. The following subtitles show the advantages of six-phase transmission line. These benefits are among the reasons why power system engineers are consistently pursues knowledge on the power system technology.73 Pthree-phase-double-circuit Because Vphase to neutral (6 phase) is hence. a six-phase system has a 73% increase in phase-to-neutral voltage. the main advantage of a six-phase transmission line is that it can carry it can carry up to 73% more electric power transfer capability compare to a threephase system at the same operating voltage. there is 73% increase in power. On the other hand.5. less conductor spacing and low insulation requirement. 2. the power capability is increased by 73%.1 Higher Power Transfer Capability Power transmission capability is directly proportional to phase-to-phase voltage. Increase in power can be evaluated as: Since. As seen by the phasor relationship. Power density refers to the amount of power that can be transmitted down a given window of ROW assuming there are environmental and technical constraints that limit size of ROW. This is especially important since ROW is becoming more difficult to obtain and increasingly expensive [12].3 Smaller Structure The phase-to-phase voltages between adjacent phases in a six-phase system are lower than the phase-to-phase voltages for a three-phase system for a given phaseto-neutral voltage. the minimum spacing between conductors on the six-phase transmission tower is reduced. The six-phase lines intrinsically have a lower likelihood of incident lightning strikes because of the smaller structure. This advantage permits smaller towers for the same power rating. The smaller structures provide increased power transfer for a given ROW. As a result. These two troubles increase the 14 . the correlation between power density and ROW is given as follow: ( ) ( ( ) ) 2.7: Determining power density Refers to Fig 2.5.Chapter 2 Six-phase Power phenomenon has already been proved in article 2. these lines can transfer more power over a given ROW than equivalently loaded three-phase lines [12]. 2.4 that power in six-phase is 1. Besides the troubles caused by the wind induced movements and visual impact can be reduced.5.2 Increased Utilization of Right-of-Way Six-phase transmission increases power density. Figure 2.73 times that of three-phase double circuit transmission line.7. Thus. 5. 2. radio and audio noise can be reduced which in turn leads to lesser television and radio interference. b) Maintain the same phase-to-neutral voltage and decrease the conductor spacing until the conductor surface electric field is a maximum for corona. 2.7 Lightning Performance When the line is converted to six-phase operation. sometimes may cause the danger of life [12]. Thus. Table 2. However. the total flashovers are so close before and after conversion that there will not be any noticeable difference in lightning performance at the line.5. 2. there is an increase in the shielding failure rate and a reduction in the back flash rate.6 Lower Corona and Field Effects Conversion from three-phase double-circuit to six-phase single-circuit has the effect of reducing electric field at the conductor surface for the same phase-to-neutral voltage.4 Lower Insulation Requirement For a six-phase system.5. Thus.Chapter 2 Six-phase Power cause of maintenance for the structures of the transmission line and which.5.5 Better Stability Margin A six-phase line can be operated at a smaller power angle than a three-phase line. Conductor gradients decrease as the number of phases increases for a given conductor size and tower configuration. [11] 15 .1 presents the results of the lightning calculation in flashovers per year referred to a line length of 2.phase counterpart [11]. The reduction in electric field can be utilized in either of two ways: a) Increase the phase-to-neutral voltage until the conductor surface electric field is a maximum for corona thus increasing the power handling capacity of the line. This means that the six-phase line offers better stability margin than its three. the insulation required to support one phase from an adjacent phase is equal to that required to support a phase from the zero potential point. utilities can save on various insulating materials for various components of transmission system [13]. thus making the line more compact.4 km. resulting in a net reduction in the trip out rate. 2. 127 2. Complexities in voltage in six-phase are discussed. six-phase have a great deal of advantages over threephase transmission system.049 Back flashes 0. For a three-phase system. Sixphase transmission system offers the opportunity to meet the increasing demands for power yet at the same time meet the environmental and regulatory constraints. Moreover. However. 120° and 180°. 2.077 Total 0. the phasors relationship can be divided into three categories. Terminal expenses can be quite high for six-phase lines. phase-to-group voltages and phase-to-cross phase voltages. the economy factors have to be considered.4 km of line [11] Configuration 115 kV three-phase 93 kV six-phase Shielding Failures 0.126 0.029 0. 16 . They are categorized depends to the phase difference between phases which is 60°. phase-to-phase voltage is equal to 3 phase-toneutral voltage.7 Summary This chapter describes the basics about six phase power and also gives an insight to its advantages and benefits. These categories are phase-to-phase voltages.155 0. In a six-phase system.6 Feasibility The aim of improving efficiency of transmission network is indeed the driving factor for electrical utility engineers to consider the six-phase transmission. The high cost of terminals is offset by reduced tower and lower foundation costs. Basic idea in six phase power transmission is introduced. A six-phase line would require conversion transformers that would cause the terminals to be more costly. ROW cost and losses. As proves that discussed in this chapter.Chapter 2 Six-phase Power Table 2.1: Goudey-Oakdale lightning performance flashovers per year for 2. The phase-to-phase voltages always lead the phase-to-neutral voltage by 30°. the phasors relationship for both three-phase and six-phase system is discussed in detail. it was first proposed as part of an international electrical committee study in 1973. of Phases The concept of using transmission systems that carry power with more than three phases is a relatively new idea.Chapter 3 Production of Six Phase Power and System Components Chapter 3 Production of Six Phase Power and System components Bulk power transmission systems in world are majorly utilizing AC transmission to transfer power do so via three phases. As stated earlier. Generating power with electrical angles less than 120 degrees between phases does not result in a corresponding increase in power output (see Fig 3.1: Machine Power Vs No. With AC power being generated at 3 phase it was logical to transfer that power in a similar manner and hence the three phase power transmission system was born. Historically this came about because threephase AC is the most efficient way to generate power. . The idea was relatively straight 17 . [12] Figure 3.1). the devices/components to be used in Six Phase power system should be analyzed. 3.2.2: Six-Phase double wye Synchronous Generator 18 .Chapter 3 Production of Six Phase Power and System Components forward. Before moving on the modeling and detailed analysis. Two major methods for the production of Six phase are: i) ii) Direct Six-phase Generation 3-phase to 6-phase conversion Detail of each method is given below. Six-phase transmission would alter the power generated into 6 phases. A double wye six-phase generator is shown in Fig 3. a double wye.1.1 Production of Six phase Six phase power can be produced in multiple ways. They may be constructed as a double delta. six-phase power can be directly generated using a six-phase synchronous generator. The construction of six phase generators may be thought of as two sets of three phase windings in the same physical housing. 3. Instead of transmitting power with the same number of phases as it was generated.1 Direct Six-phase Generation As discussed earlier. This process would allow for some unique benefits that are described in previous chapter. In this chapter. the methods of production of Six phase power and components used Six phase power Transmission system are analyzed that include six phase generator. However. six phase transformer and six phase transmission line. [14] Figure 3. the generation in three-phase is the most efficient way to generate electric power [12]. or one wye plus one delta. The term power transformer is used to refer to those transformers used between the generator and the distribution circuits and are usually rated at 500 kVA and above. 3. Following article deals with the Power Transformer to be used in Six-phase Transmission. six identical single phase two winding transformers may be connected to form three to six-phase transformer bank. and interconnections within the system or with nearby systems.2 Three-phase to Six-phase conversion The other and most feasible method for the production of six phase is by using three phase to six phase conversion transformer bank.2 Power Transformer A transformer is defined as a static electrical device. involving no continuously moving parts. Secondly. The details about transformer connections and their characteristics are discussed in the next articles.1.Chapter 3 Production of Six Phase Power and System Components However. three identical single phase three winding transformers may be connected together to form three to six-phase transformer bank. Since electric power is proportional to the product of voltage and current. First. Power transformers are selected based on the application. It finds its application in speed control of drives and renewable energy generation. Voltage and current magnitude depends on the windings connections. It establishes the definition of transformer. with the emphasis towards custom design being more apparent the larger the unit. 3. primarily 19 . A six-phase to three-phase or three-phase to six-phase conversion transformer can be constructed by two techniques. Power systems typically consist of a large number of generation locations. such as a neighboring utility. Power transformers are available for step-up operation. used in electric power systems to transfer power between circuits through the use of electromagnetic induction. low current levels (and therefore low I²R losses and low IZ voltage drops) can be maintained for given power levels at the expense of high voltages. distribution points. discussed three-phase transformer and then leads to the six-phase transformer and its connection. six-phase generator does not have its practical applications for power generation in bulk due to its higher complexity and less efficiency than three-phase generation machines. The power transformer is a major power system component that permits economic power transmission with high efficiency and low series-voltage drops. so the voltage is more accurately described (3. the voltages on the input and the output are related by the turn’s ratio of the transformer and given as below: (3. In an ideal transformer. This leakage flux creates a voltage drop between windings. The construction of a transformer depends upon the application. Each unit in a bank should have the same voltage ratings but need not supply the same kVA load.2) The current also transforms by the turns ratio. mainly used to feed distribution circuits. Power transformers are available as a single-phase or three-phase apparatus.3. not all of the flux couples between windings. The example of outdoor liquidimmersed transformers has been shown in Fig 3. 20 . opposite of the voltage as (3.3: 20 MVA three-phase transformers A transformer is two sets of coils coupled together through a magnetic field.3) Single-phase transformers can be connected into banks of two or three separate units. Figure 3. with transformers intended for indoor use primarily dry-type but also as liquid immersed transformers are used and for outdoor use usually liquid immersed transformers are used.Chapter 3 Production of Six Phase Power and System Components used at the generator and referred to as step-up transformers (SUT).1) In a real transformer. The primary winding of a single-phase transformer can be connected between a phase conductor and ground or between two phase conductors of the primary system. and for stepdown operation. Wye-Wye. The kVA rating for a three-phase bank is the total of all three phases.Chapter 3 Production of Six Phase Power and System Components 3. Secondly. This is especially true of three-phase transformers using common core structures. One end of each primary lead has been labeled as H1 and the other 21 . Stacked cores have three or possibly four vertical legs. Y-Y.4 have been labeled as A. Threephase transmission line terminal transformer services are normally constructed from three single. There are four common combinations used in three-phase transformer which is.3.4) There are two ways that can be used to construct a three-phase transformer. ∆-Y. Three-phase transformers for underground service (either pad mounted. i. Voltage and current magnitude depends on the windings connection used at the primary and the secondary sides of that three-phase transformer. iv.1 Y-Y Connection The three transformer windings in Fig 3. First. iii. The use of three versus four or five legs in the core structure has a bearing on which electrical connections and loads can be used by a particular transformer. Y-∆.or five-legged core. usually on a three. The primary or secondary sides of the three-phase transformer may be connected by using either Wye (Y) or Delta (∆) connections. three identical single-phase two-winding transformers may be connected to form three-phase bank. while wound cores have a total of four loops creating five legs or vertical paths: three down through the center of the three coils and one on the end of each outside coil. (Y-Y) Wye-Delta (Y-∆) Delta-Wye (∆-Y) Delta-Delta (∆-∆) 3. B and C respectively.phase units. direct buried. The full-load current in amps in each phase of a three-phase unit or bank is: √ (3. The advantage of three-phase electrical systems in general is the economy gained by having the phases share common conductors and other components. ii. ∆-∆. a three-phase transformer can be constructed by winding three single-phase transformers on a single core.3 Three-Phase Transformer Connections Three-phase transformers have one coaxial coil for each phase encircling a vertical leg of the core structure. or in a vault or building or manhole) are normally single units. The schematic diagram for Y-Y connected threephase transformer is shown in Fig 3.2 Y-∆ Connection Fig 3.7.6 shows the three-phase transformer with Y-∆ connection.Chapter 3 Production of Six Phase Power and System Components end has been labeled as H2.4: Y-Y connected three-phase transformer Figure 3. The three transformer windings have been connected to form a three-phase transformer with Y-Y connection as shown in Fig 3.3. Turn ratio of a transformer is generally written as ‘a’.5. Furthermore. √ √ (3. voltage relation on primary winding for all phase is given by VPP = VLP/√3. Primary phase-to-neutral voltage relates to secondary phase-to-neutral voltage by number of winding turns. The schematic diagram for Y-∆ connected three-phase transformer is shown in Fig 3. For a three-phase transformer with Y-Y connection.4. one end of each secondary lead has been labeled as X1 and the other end has been labeled as X2.5) Figure 3.5: Schematic diagram of Y-Y connected three-phase transformer 3. The relation of phase-to-neutral voltage and phase-to-phase voltage on secondary side is given by VLS =√3VPS [3]. The relation between phase-to-neutral voltage and phase-to-phase voltage for primary and secondary side is given by [3]: 22 . 3.Chapter 3 Production of Six Phase Power and System Components VLP =√3VPP & VLS =VPS √ √ (3.6: Y-∆ connected three-phase transformer Figure 3.3 ∆-Y Connection Fig 3.8 shows the three-phase transformer with ∆-Y connection.6) Figure 3.9.3. The schematic diagram for ∆-Y connected three-phase transformer is shown in Fig 3. and secondary side is given by [3]: VLP =VPP & VLS =√3VPS The relation between phase-to-neutral voltage and phase-to-phase voltage for primary √ 23 .7: Schematic diagram of Y-∆ connected three-phase transformer. 8) 24 .9: Schematic diagram of ∆-Y connected three-phase transformer 3.8: ∆-Y connected three-phase transformer Figure 3.4 ∆-∆ Connection Fig 3.Chapter 3 Production of Six Phase Power and System Components √ (3. The relation between phase-to-neutral voltage and phase-to-phase voltage for primary and secondary side is given by [3]: VLP = VPP VLS = VPS (3. The schematic diagram for ∆-∆ connected three-phase transformer is shown in Fig 3.7) Figure 3.10 shows the three-phase transformer with ∆-∆ connection.11.3. Voltage and current magnitude depends on the windings connection used on the primary and the secondary sides of the three-to-six-phase conversion transformer. Diametrical. 3.1 Y-Y and Y-Inverted Y The six transformer windings in Fig 3. three identical single-phase three-winding transformers may be connected together to form three-to-six-phase bank. C. Double-Delta and Double. The primary or secondary side of the three-to-six-phase conversion transformer may be connected by using any combinations of either Wye (Y) or Delta (∆) connections.4 Six-Phase Transformer Connections As discussed earlier.10: ∆-∆ connected three-phase transformer Figure 3.Chapter 3 Production of Six Phase Power and System Components Figure 3. First. D. B. one end of each secondary lead has been 25 .11: Schematic diagram of ∆-∆ connected three-phase transformer. Secondly. there are two types of single-phase transformers that can be used to build a three-to-six-phase conversion transformer. six identical singlephase two-winding transformers may be connected to form three-to-six-phase bank. ∆-Y & ∆-Inverted Y. There are five common connections and combinations that can be used to form a three-to-six-phase conversion transformer which is Y-Y and Y-Inverted Y.12 have been labeled as A. E and F respectively. One end of each primary lead has been labeled as H1 and the other end has been labeled as H2.Wye. 3. Furthermore.4. 13. At the other hand. L4. L5 and L6. Neutral line name as N is the common for all neutral lines of transformers. Fig 3. L4 and L6.Chapter 3 Production of Six Phase Power and System Components labeled as X1 and the other end has been labeled as X2. we can see that the first three-phase transformer is used Y-Y connection and produced three phase line on the secondary side name as lines L1. Figure 3.9) That means in Y-Y and Y-Inverted Y connection. the second three-phase transformer is used Y-Inverted Y connection and produced another three phase line on the secondary side name as lines L2. From Fig 3. The schematic diagram for Y-Y and Y-Inverted Y connected three-to-six-phase conversion transformer is shown in Fig 3. line voltage decreases and becomes equal to phase voltage.12 and Fig 3. L2. L3.13.12 shows the Y-Y and Y-Inverted Y connected three-to-six-phase conversion transformer. Combination of all these line will produce six-phase line name as L1. VLP = √ VPP & VLS =VPS √ √ (3. L3 and L5. on secondary side.12: Y-Y and Y-Inverted Y connected three-to-six-phase conversion Transformer 26 . 27 . The schematic diagram for ∆-Y and ∆-Inverted Y connected three-tosix-phase conversion transformer is shown in Fig 3.13: Schematic diagram of Y-Y and Y-Inverted Y connected three-to-six.14: ∆-Y and ∆-Inverted Y connected three-to-six-phase conversion Transformer.10) That means in ∆-Y and ∆-Inverted Y connection.phase conversion transformer. on secondary side.4. VLP = VPP & VLS = VPS (3. phase voltage increases and becomes equal to line voltage.Chapter 3 Production of Six Phase Power and System Components Figure 3.14 shows the ∆-Y and ∆-Inverted Y connected three-to-six-phase conversion transformer. 3.15.2 ∆-Y and ∆-Inverted Y Fig 3. Figure 3. 3 Diametrical Fig 3. The schematic diagram for Diametrical connected three-to-six-phase conversion transformer is shown in Fig 3.Chapter 3 Production of Six Phase Power and System Components Figure 3. 3.16: Diametrical connected three-to-six-phase conversion transformer VLP = VPP & VLS = VPS (3.4.11) 28 .17.phase conversion transformer. Figure 3.15: Schematic diagram of ∆-Y and ∆-Inverted Y connected three-to-six.16 shows the Diametrical connected three-to-six-phase conversion transformer. Figure 3. The schematic diagram for Double-Delta connected three-to-six-phase conversion transformer is shown in Fig 3.19. phase voltage increases and becomes equal to line voltage.18: Double-Delta connected three-to-six-phase conversion transformer VLP = √ VPP & VLS =VPS √ 29 . 3. on secondary side.18 shows the Double-Delta connected three-to-six-phase conversion transformer.17: Schematic diagram of Diametrical connected three-to-six-phase conversion transformer.4. Figure 3.Chapter 3 Production of Six Phase Power and System Components That means in diametrical connection.4 Double-Delta Fig 3. 19: Schematic diagram of Double-Delta connected three-to-six-phase conversion transformer.12) That means in Double Delta connection. 3. The schematic diagram for Double-Wye connected three-to-six-phase conversion transformer is shown in Fig 3.Chapter 3 Production of Six Phase Power and System Components √ (3. Figure 3. on secondary side. VLP = VPP & VLS = VPS 30 .20 shows the Double-Wye connected three-to-six-phase conversion transformer.21.20: Double-Wye connected three-to-six-phase conversion transformer. Figure 3. line voltage decreases and becomes equal to phase voltage.4.5 Double-Wye Fig 3. Pakistan is one of the Asian countries that use 50 Hz as the operating frequency. 31 . The three-phase system has three phase conductors while six-phase system has six phase conductors. The overhead transmission lines are used in open areas such as interconnections between cities or along wide roads within the city. and 50 Hz in Europe. on secondary side.S. The underground transmission system is environmentally preferable but has a significantly higher cost. Australia.13) That means in Double-wye connection. and part of Asia.21: Schematic diagram of Double-Wye connected three-to-six-phase conversion transformer. The operating frequency is 60 Hz in the U. Transmission lines also interconnect neighboring utilities which permits not only economic dispatch of power within regions during normal conditions. Figure 3. 220 kV. The cost per mile of overhead transmission lines is 6% to 10% less than underground cables [3]. 500 kV and 765 kV line-to line. Transmission voltage lines operating for NTDC system are standardized at 132 kV.Chapter 3 Production of Six Phase Power and System Components (3. underground cables are used for electric energy transmission. phase voltage increases and becomes equal to line voltage.5 Power Transmission Line The purpose of an overhead transmission network is to transfer electric energy from generating units at various locations to the distribution system which ultimately supplies the load. Transmission voltages above 220 kV are usually referred to as extra-high voltage (EHV). but also the transfer of power between regions during emergencies. 3. In congested areas within cities. Standard transmission voltages are established in by NTDC in Pakistan. these will give the impedance and admittance as follows [3]: z = jωL Ω/m y = jωC Ω/m Characteristic impedance Zc is given by [3]: (3. Moreover.1 Surge Impedance System limitations on power flow include among other considerations voltage drop and stability.Chapter 3 Production of Six Phase Power and System Components A three-phase double-circuit AC system is used for most transmission lines. A three-phase double-circuit transmission line can be easily converted to a six-phase transmission line by using three-to-six-phase conversion transformer.14) (3. For the reason of this fact. Since transmission and distribution lines for power transfer generally are designed to have low losses. When line losses are neglected. 3. One of each pair of transformers has reverse polarity to obtain the required 60° phase shift. This combination were selected as appropriate for determining short circuit currents because the delta open circuits the zero sequence network and simplifies the fault analysis [15]. the most suitable one is by using two pairs of identical delta-wye three-phase transformers. There are several combinations of identical three-phase transformers that can be used to form three-to-six-phase and six-to-three-phase conversion transformers. For a lossless line.5. R = G = 0. This will make the idea of transmitting power using six-phase transmission system much easier because six conductors of three-phase double-circuit transmission line can be converted to six-phase transmission line. the equations and concepts shows here can be used for quick and reasonably accurate hand calculations leading to initial designs. More accurate calculations can then be made with computer programs for follow-up analysis and design.15) 32 . This section will discuss the concepts of surge impedance and surge impedance loading for lossless lines. However. three-tosix. Conversion of an existing three-phase double-circuit overhead transmission line to a six-phase operation needed phase conversion transformers to obtain the 60° phase shift between adjacent phases. A rule of thumb estimate of power-handling capacity of a transmission line is given by line-surge-impedance loading.phase and six-to-three-phase conversion transformers that forms by using this combination has been used throughout this study. simpler expression for the line parameters are obtained and above concepts are more easily understood. resistive.3 Line Loadability In practice.Chapter 3 Production of Six Phase Power and System Components √ √ √ √ ⁄ (3. This line represents either a single-phase line or one phase-to-neutral of balanced threephase or six-phase line. an increase in voltage can have a significant impact on the line SIL. it is a useful way of visualizing the impact of a conversion which allows an increase of line voltage with small change in surge impedance.22 shows a lossless line terminated by a resistance equal to its surge impedance.and six-phase transmission line alternatives. it is possible to develop six-phase lines with comparable thermal or surge-impedance loading characteristics to the three33 . power line are not operated to deliver their theoretical maximum power. Because SIL is a function of square of the phase-to-neutral voltage. The reactive power flow is zero. The relation of line loadability to SIL as a function of line length is given in Fig 3.5. is pure real-that is. When developing three.23. At rated line voltage.16) is commonly called surge impedance for The characteristic impedance a lossless line. SIL is that loading at which VARs generated in the line capacitance cancel the VARs absorbed in the line inductance and is equivalent to the case of an impedance-matched line.22: Lossless line terminated by its surge impedance. The real power along the lossless line at SIL remains constant from the sending end to the receiving end.5. which is based on rated terminal voltages and an angular displacement δ = 90° across the line. While a transmission system would not be constructed according to the curve in Fig 3.23. the real power delivered (SIL) is given by [16]: (3.2 Surge Impedance Loading Surge Impedance Loading (SIL) is the power delivered by a lossless line to load resistance equal to the surge impedance √ ⁄ . 3. Fig 3. Figure 3.17) 3. this statement is somewhat of an over simplification. The appropriate comparison to use is related to the specific application. thus generally enhancing system stability in the same manner as system stability is enhanced by any conversion that results in a higher line operating voltage. 34 . If the angle δ exceeds 90°.4 Stability Performance The power flow through any transmission line. especially whether the line limits the system or the system limits the line.23: Surge impedance loading characteristic curve [16] 3. the power decreases with increasing angle. neglecting the effect of line resistance is given by [16]: (3.18) The power flow is maximum when δ = 90°. Figure 3. because there is additional margin for the system to swing without exceeding the 90° limit. a condition of voltage instability. The basic effect of a six-phase line on system stability is similar to the effect of a higher-voltage three phase line and must be evaluated by the same type of stability analysis. because a higher-voltage line generally carries a greater load.5. Of course.Chapter 3 Production of Six Phase Power and System Components phase alternative. System changes which reduce δ for the same power enhanced the system stability. Increasing phase-to-neutral voltage by a six-phase conversion increases the per-unit positive-sequence impedance. which results in a greater system disturbance in the event of a line trip. transmission line have been viewed.Chapter 3 Production of Six Phase Power and System Components 3. the theoretical aspects involved in the conversion of a three-phase transmission line to a six-phase 35 . Methods of production of six-phase have been discussed followed by a detailed analysis of three to six-phase conversion transformers.6 Summary This chapter deal with the components in involved in Six-phase transmission system. Voltage relationships of primary (three-phase side) and secondary (six-phase side) of the conversion transformers are also derived. In the end. 2 SimPowerSystems SimPowerSystems software is a modern design tool that allows scientists and engineers to rapidly and easily build models that simulate power systems. steam. and other disciplines. Engineers working in this discipline are constantly improving the performance of the systems. but your analysis of the circuit can include its interactions with mechanical. A common attribute of these systems is their use of power electronics and control systems to achieve their performance objectives.Chapter 4 Modeling of six-phase Transmission System in MATLAB® Chapter 4 Modeling of six-phase Transmission System in MATLAB® 4. Land-based power generation from hydroelectric. This is possible because all the electrical parts of the simulation interact with the extensive Simulink modeling library. thermal. control. allowing you to build a model using simple click and drag procedures. or other devices is not the only use of power systems.1 The Role of Simulation in Design Electrical power systems are combinations of electrical circuits and electromechanical devices like motors and generators. Not only can you draw the circuit topology rapidly. It uses the Simulink environment. designers can also use MATLAB® 36 . Further complicating the analyst's role is the fact that the system is often so nonlinear that the only way to understand it is through simulation. Since Simulink uses the MATLAB® computational engine. Requirements for drastically increased efficiency have forced power system designers to use power electronic devices and sophisticated control system concepts that tax traditional analysis tools and techniques. 4. and consumption of electrical power. distribution. electric drives. 4. configure.1 Simulation and Model-Based Design Simulink® is an environment for multi domain simulation and Model-Based Design for dynamic and embedded systems. flexible AC transmission systems (FACTS). You can use these components to model the generation. Following are the Key Features of Simulink®:        Extensive and expandable libraries of predefined blocks Interactive graphical editor for assembling and managing intuitive block diagrams Ability to manage complex designs by segmenting models into hierarchies of design components Model Explorer to navigate. and generated code associated with your model Application programming interfaces (APIs) that let you connect with other simulation programs and incorporate hand-written code MATLAB® Function blocks for bringing MATLAB algorithms into Simulink and embedded system implementations Graphical debugger and profiler to examine simulation results and then diagnose performance and unexpected behavior in your design 4. 37 . parameters. Abstracted models of power electronics components are also included. implement. create.2. video processing. including communications.Chapter 4 Modeling of six-phase Transmission System in MATLAB® toolboxes and Simulink block-sets. and wind power generators. and image processing. It provides an interactive graphical environment and a customizable set of block libraries that let you design. transmission. and test a variety of time-varying systems. simulate. properties. signal processing. controls.2.2 Model and simulate electrical power systems SimPowerSystems provides component libraries for modeling and simulating electrical power systems. SimPowerSystems software belongs to the Physical Modeling product family and uses similar block and connection line interface. It includes models of three-phase machines. and search all signals. enabling you to assess the impact of switching events on system-level behavior. The powerlib library window displays the block library icons and names. there are also self-learning case studies. and other key electrical power system analyses are automated. 38 . including common AC and DC electric drives. organizes its blocks into libraries according to their behavior. type powerlib in the MATLAB® Command Window. and wind-power generators Ideal switching algorithm for fast simulation of power electronic devices Functions for obtaining equivalent state-space representations of circuits Tools for computing load flow and for initializing models of three-phase networks with machines Demonstration models of key electrical technologies 4. SimPowerSystems models can be discretized to speed up simulations and configured for phasor simulation. calculation of total harmonic distortion (THD). and their validity is based on the experience of the Power Systems Testing and Simulation Laboratory of Hydro-Québec. and also on the experience of École de Technologie Supérieure and Université Laval. Key Features of SimPowerSystems are [17]:      Application-specific models. Double-click a library icon to open the library and access the blocks. And for users who want to refresh their knowledge of power system theory. The main powerlib library window also contains the Powergui block that opens a graphical user interface for the steady-state analysis of electrical circuits.Chapter 4 Modeling of six-phase Transmission System in MATLAB® Harmonic analysis. To open this library. which helps you determine the transient stability of electrical power systems. These models are proven ones coming from textbooks. machines. and power electronics. powerlib.3 Overview of SimPowerSystems Libraries SimPowerSystems libraries contain models of typical power equipment such as transformers. The capabilities of SimPowerSystems software for modeling a typical electrical system are illustrated in demonstration files. a large North American utility located in Canada. flexible AC transmission systems. [17] The SimPowerSystems main library. load flow. lines. Chapter 4 Modeling of six-phase Transmission System in MATLAB® Figure 4.1 Nonlinear Simulink Blocks for SimPowerSystems Models The nonlinear Simulink blocks of the powerlib library are stored in a special block library named powerlib_models. To access Block Libraries you can also access SimPowerSystems libraries through the Simulink Library Browser. type Display in tab and press ENTER. the following window will appear 39 .2 Simulink Library Browser To search any block type the name of block in Searching Tab e. you can type simulink in the MATLAB Command Window. click the Library Browser button in the toolbar of the MATLAB desktop or Simulink model window: Alternatively. Figure 4. Then expand the Simscape entry in the contents tree. To display the Library Browser. These masked Simulink models are used by SimPowerSystems software to build the equivalent Simulink model of your circuit.g. up to 500kV and also converting it to double circuit line by using transformation block as shown in fig.5 shows the internal connections of the transformation block.4. Fig.4 Modeling of Three-phase double circuit line on Simulink First of all.4. 4. a model of three-phase double circuit line was built in Simulink.4.3 Display block for numeric display of input values Select your desired element and connect it in the system.Chapter 4 Modeling of six-phase Transmission System in MATLAB® Figure 4. A three phase source at voltage level of 220kV was taken and then it was stepped up. Figure 4-4 Block diagram of Three phase transformer 40 . 4-6.5 Block Diagram and Connection Diagram of Three Phase T/F Then power is transferred towards load by using two circuits of transmission lines as shown in fig. with the only difference that primary and secondary connections are interchanged. Figure 4. The wave shapes of voltages are shown in fig.Chapter 4 Modeling of six-phase Transmission System in MATLAB® Figure 4. 41 .6 Transmission Line Again using a transformation block similar to that used for step up is used for step down purpose. 4-7. 8 Waveform of Line Voltages These graphs show that line voltages are √ times greater than the phase voltages. 42 .7 Waveform of Phase Voltages Figure 4.Chapter 4 Modeling of six-phase Transmission System in MATLAB® Figure 4. The block diagram and internal connections of phase voltage and line voltage are shown respectively. Two measurement blocks are formed for phase voltage measurements and line voltage measurements separately. 9 Hierarchy of Measurement blocks for Phase Voltages Figure 4.10 Hierarchy of Measurement blocks for Line Voltages 43 .Chapter 4 Modeling of six-phase Transmission System in MATLAB® Figure 4. 12 Three-Phase RLC load 4. For this modal of power system a Three-Phase RLC load of following settings is used Figure 4.11 Complete model of Three Phase double circuit Transmission System The line current is almost 156A.Chapter 4 Modeling of six-phase Transmission System in MATLAB® Figure 4.5 Modeling of six phase transmission system 44 . e. at an angle of 60°.1 [18]. There are different transformer configurations used for this type of conversion and these configurations are shown in the table 4. Type of Connection Wye-Wye Wye-Inverted-Y Schematic Diagram Delta-Wye Delta Inverted-Y Diametrical 45 .Chapter 4 Modeling of six-phase Transmission System in MATLAB® Power system model for six phase transmission lines is similar to that shown in figure 4-10 but the only difference here is that connections of transformers in transformation blocks are such that three phase voltages of source are stepped up and are also converted into six phase and all the phases are equally displaced from each other i. 5. 4-13 Figure 4.1 Transformation block for wye-wye wye-inverted-wye The internal configuration of six single phase transformers in transformation block of power system is shown in fig.13 Y-Y Y-Inverted Y Configuration of Transformers 46 .1 4.Chapter 4 Double Delta Modeling of six-phase Transmission System in MATLAB® Double Wye Table 4. The wave shapes of voltages are shown in graphs of fig. 4-15. Figure 4. 4-14 and fig.Chapter 4 Modeling of six-phase Transmission System in MATLAB® Simulation results show that in this case phase voltage is not increased to line voltages but line voltage is decreased to phase voltages. Hence this configuration cannot be used for enhancement of power capability of transmission lines.14 Waveform of Phase voltages Figure 4.15 Waveform of Line Voltages 47 . Source voltages or voltages before the conversion blocks / transformation blocks the three phase voltages are shown in fig. we can conclude that magnitude of line voltage is reduced to the phase voltage in case of six phase line. The above graphs also clearly indicate that magnitude of phase voltages are approx. 4-16.16 Source Voltages After the conversion of six phases back into three phases the wave shapes are shown in fig. equal to line voltages and phase difference between two consecutive phases is 60°. 4-17 48 .Chapter 4 Modeling of six-phase Transmission System in MATLAB® By comparing these graphs with the graphs of three phase double circuit graphs. Figure 4. 17 Voltages across Load The overall power system modeling is shown in fig. Figure 4-18 Complete System for Six Phase Transmission Using Y-Y. Y-Inverted Y Transformer configuration 49 . 4-18.Chapter 4 Modeling of six-phase Transmission System in MATLAB® Figure 4. The results of simulation are shown in fig. 50 . 4-20 and 4-21. In this configuration phase voltage is increased to the line voltage in contrast to the wye-wye wye-inverted wye configuration where line voltage is reduced to the phase voltage. Figure 4. So in this configuration power transfer capability of transmission line is also increased up to √ times.5.19 Hierarchy of Delta-Wye Delta-Inverted Wye Transformation block Voltage phasors can be viewed by connecting scopes at line and phase measurement blocks.Chapter 4 Modeling of six-phase Transmission System in MATLAB® 4.2 Delta-wye Delta-Inverted wye configuration for conversion into six phase transmission The internal connections of transformation / conversion blocks are as shown below for delta-wye delta-inverted wye configurations. Assuming the VAN = VBN = VCN = VDN = VEN = VFN =VP.20 Waveform of Phase Voltages Figure 4.Chapter 4 Modeling of six-phase Transmission System in MATLAB® Figure 4. Magnitude of line voltage is equal to phase voltage in magnitude but line voltage leads the phase voltage by 60°. It is theoretically can be verified as. Phase-to-phase voltage is a potential between adjacent phases where their phase difference is 60°.21 Waveform of Line Voltages If we carefully examine the above two graphs it would be clear that the phase voltage has increased to line voltage. 51 . Chapter 4 Modeling of six-phase Transmission System in MATLAB® VAB = VAN ∠ 0°- VBN ∠ -60° = VP (1 ∠ 0°- 1 ∠ -60°) = VP (1+j0 - (0.5 - j0.866)) = VP (0.5 + j0.866) = VP ∠ 60° Using this configuration 73 % extra loading of transmission lines is permissible. So, for the same current in line power of RLC series load is multiplied by 1.73 i.e. active, inductive, capacitive powers are multiplied by factor of 1.73. So, capability increases √ times but current in a conductor is almost same as it was in three phase double circuit (157A). Settings of RLC load are shown in Fig. 4-22. Figure 4- 22 Three-Phase RLC Load The whole power system is shown in Fig. 4-23. 52 Chapter 4 Modeling of six-phase Transmission System in MATLAB® Figure 4-23 Six Phase Transmission System using Delta-Wye Delta-Inverted Wye Configuration of Transformer 4.5.3 Diametrical configuration for six phase Transmission Following fig shows schematic diagram for diametrical configuration of phase conversion. Figure 4-24 Diametrical Configurations 53 Chapter 4 Modeling of six-phase Transmission System in MATLAB® In Simulink we used same power transmission system shown in fig. 4-22 with the only difference that transformation blocks are replace with the following blocks; Figure 4-25 Block Diagrams The internal structure of the above blocks is shown in fig. 4-26; Figure 4-26 The connection diagram of Diametrical conversion transformer 54 6 Voltage Drop Comparison In this section we compare the voltage drops of six-phase transmission system with that of three-phase double circuit transmission line.Chapter 4 Modeling of six-phase Transmission System in MATLAB® The results of diametrical configurations are exactly same as that of delta-wye delta-inverted-wye configurations. Following parameters are entered for both of three-phase double circuit and six-phase transmission lines.2.5. shown in section 4. 4.3. Line length is taken to be 30km.2 lists the voltage drops along the length of transmission line for a ThreePhase Double Circuit (TPDC) Transmission Line (T.L). 55 . Line Parameters entered here are in per unit and are those obtained from National Transmission and Dispatch Company’s (NTDC) for a nominal 500kV transmission line. Figure 4-27: Input Parameters of Transmission Line Table 4. This line is converted to Six-Phase Single Circuit (SPSC) transmission line and the voltage drops across a length of 30km transmission line are observed and also shown in Table 4. This table verifies that the voltage drop across the six-phase transmission line is greater than that of three-phase double circuit transmission line that is a demerit of six phase transmission line. 1 2 3 4 5 6 Phase A/A B/B C/C D/A’ E/B’ F/C’ SPSC T. 56 . Table 4.e 1.L Voltage Drop 1249 Vrms 1249 Vrms 1249 Vrms 1249 Vrms 1249 Vrms 1249 Vrms It is quite easily observable that the voltage drops in six-phase transmission line are greater than those in three-phase double circuit transmission line.L Voltage Drop 1862 Vrms 2366 Vrms 2261 Vrms 1862 Vrms 2366 Vrms 2261 Vrms TPDC T.2: Voltage Drop across the length of transmission lines for three-phase and six-phase with 73% extra load.73 times that of three phase double circuit transmission line. Quantity of power flow or power transfer capability and voltage drops for both the transmission line is compared.3 lists the voltage drops across a 30km long 500kV three phase double circuit and six-phase transmission line for the same load. This may be due to the increased load i.L Voltage Drop 1249 Vrms 1249 Vrms 1249 Vrms 1249 Vrms 1249 Vrms 1249 Vrms So we note here that the six-phase power transmission line has poor voltage regulation than that of three-phase double circuit transmission line. To verify. Sr. 4.3: Voltage drops across the length of transmission line for Six phase with same load as threephase. Sr. we have again listed the voltage drops in six-phase transmission line that are due the line with same load as that of three-phase double circuit line. No.7 Summary In this chapter modeling and comparison of three-phase double circuit and sixphase single circuit transmission lines are performed in Simulink/MATLAB®.Chapter 4 Modeling of six-phase Transmission System in MATLAB® Table 4.L Voltage Drop 3342 Vrms 4177 Vrms 3961 Vrms 3342 Vrms 4177 Vrms 3961 Vrms TPDC T. No. Table 4. 1 2 3 4 5 6 Phase A/A B/B C/C D/A’ E/B’ F/C’ SPSC T. Both of these elements directly depend on the electric and magnetic field gradients around a transmission line. A MATLAB program has been developed for the calculation and plotting of Six-phase transmission line electric and magnetic field.1 Magnetic Field Basics An electric charge has an electric field.Chapter 5 Electromagnetic Field Gradients Chapter 5 Electromagnetic Field Gradients In the recent era the. In the calculation 57 . Moreover. Basic definitions and equations are described first followed by the case study of six-phase transmission line. One of the major advantages of six-phase transmission is less ROW requirement. the construction of new six-phase transmission line using compact structures requires less land for its construction. ROW requirement directly depends upon the size of tower structures and electromagnetic field limits imposed by the Environment Safety Authorities. Now we start from calculating the magnetic field. 5. The up-gradation of existing three-phase double circuit transmission line to sixphase transmission line eliminates the construction of new line and hence ROW requirement. In this chapter we analyze the Electric and Magnetic field across a Six-phase transmission line and compare it with the three-phase double circuit transmission line under same tower structures. construction of new electrical power transmission lines are strongly constrained by the Right of Way Requirement. Increasing cost of land and legal issues involved in the acquiring land have compelled electric design engineers to look for alternatives to transmit power to the distribution stations. The magnetic field is considered as static in case of DC transmission and quasi-static in the case of AC transmission lines. while an electric current produces a magnetic field. Biot-Savart Law states that the differential magnetic field strength is independent of the medium and is expressed in vector notationas shown in Fig 5. current carrying conductor The conductor is positioned along the Z-axis and carries a current I. there is no H variation with Z as shown in Fig 5. i.1: (5. Ampere’s Law states that the line integral of H about any closed path is exact equal to the current enclosed by the path.Chapter 5 Electromagnetic Field Gradients of magnetic field for transmission lines.2 Application of Ampere’s Law to infinitely long. The contour may close at infinity. 5.2) The closed line integral simply requires that all current elements be included in order to obtain the complete H. Current elements have no separate existence.1 Basic Concepts: A conductor carrying a current I has a magnetic field surrounding it. some assumptions are involved.1) Figure 5-1: The BiotSavart Law The distance R is from the center of the current element to the point at which dH is to be determined. The relation of the magnetic field direction to the current direction can be determined by means of the right-hand rule. All elements making up the complete current filament contribute to H. (5.e. Using the Biot-Savartlaw it is possible to conclude that the direction of dH is perpendicular to the plane 58 .3) In order to use Ampere’s law to obtain H there must be considerable degree of 5. By symmetry inspection. and must be included. ∮ symmetry in the problem.1. the current flows in Z direction in a cylindrical coordinate system.2.1. The summation leads to the integral form of the Biot-Savart law [19]: ∮ (5. First we establish the basics by calculating a magnetic field for a single conductor line. the single conductor case is reviewed as shown in Fig 5.3.Chapter 5 Electromagnetic Field Gradients [20]. The Ampere’s law becomes: (5. therefore for power frequency (50 Hz) currents.3 Application to Transmission Lines First.1. the conductor depth is very large. (5. about 1 km. Hence the only containing dL and R and hence is in the direction of component of H is . Thus. the depth of the equivalent conductor is given by: √ ⁄ Where.5) The soil resistivity is usually in the order of 100 or 150 ohm-meters and. The results obtained are extended for the multi-conductor case. the influence of the return conductors through ground can be neglected in practical magnetic field calculations. 5. and it is only a function of r radius. integration is done along the circle of radius r.3: Magnetic Field of a single conductor 59 . Figure 5. Figure 5.4) In case of unbalanced faults to ground or unbalanced loads with return through ground.2: Magnetic field of aconductor along Z-axis carrying current I To simplify the integral form. 5.yp) to have a graphical picture of the magnetic field.4 can be used: | | √ | | | | (5. then Eq.8) (5.14) Eq. current magnitudes with phasors and the space (set of points) in which the magnetic field is to be evaluated.[19] Figure 5. i represents the conductor number and Iirepresents the current in conductor i as a phasor as shown in Fig 5.14 gives the magnitude of the electric field strength vector.6) (5.4 Computer Program for calculation of Magnetic Fields A program in MATLAB is written for the calculation of magnetic field.Chapter 5 Electromagnetic Field Gradients If current I is given as a phasor.7) (5.1. The output of the program is the value of magnetic field strength. 5. 5. This equation can be plotted against the points (xp. The input to the program is the geometry of the tower.11) ∑ ∑ (5.13) (5. a plot of magnetic field 60 .4.12) ∑ | | √ ∑ (5.4: Magnetic field of a multi-conductor line ∑ ∑ √ (5.10) (5.9) In case of a multi-conductor line. y2).(x6. (x2. Figure 5. This function is called again and again for 501x501 points to evaluate magnetic field.Chapter 5 Electromagnetic Field Gradients strength versus distance from transmission line and a complete profile of transmission line magnetic field strength. y6). One of the functions is called by the other function gives magnetic field strength against a single point input in the form of x and y coordinates of the point. Tower geometry to the program is given by the six points. (x1. The program is divided into two functions. The resultant matrix is then plotted against the x and y arrays to form a magnetic field profile in the form of contours.. y1).5. Tower geometry representing these points is given in Fig 5.5: Relation between the lengths and Tower Geometry 61 .…. 2.38 Current I (A) 1000 1000 1000 1000 1000 1000 Phasor (Degrees) 0o 120o 240o 0o 240o 120o The overall magnetic field profile drawn by the program for three-phase double circuit transmission line is given in Fig 5.1: Input data for three-phase Double Circuit Transmission Line Sr. We start with the plotting the magnetic field strength for three phase double circuit transmission line and plotting them.Chapter 5 The computer Program is listed in Appendix A.1 6.86 21. The current in a single conductor is 1000A.1. No.34 21.1 6.1 6.1 Magnetic Field of Three-Phase Double Circuit Line Here we suppose an infinite 220kV three-phase double circuit transmission line delivering a total load of 1320MVA.2 Magnetic field strength for Six-phase Line Here we present the plots of six-phase transmission line and compare them with those of three phase double circuit transmission line.86 24.6: Magnetic Field Profile of Three-phase Double Circuit Transmission Line 62 .1 Yi (m) 24. Table 5. The input data to the computer program is given in Table 5. Figure 5.34 22. 1 2 3 4 5 6 Xi (m) 6.38 22. Electromagnetic Field Gradients 5.1 6.6. 5.1 6. 1 6.7: Magnetic field of three-phase double circuit transmission line. The input data to the computer program is given in Table 5.2.34 21.2: Input Data for Six-phase line with same load Sr.1 6.86 24. the plot of magnetic field strength versus the distance along a slope of 2 is plotted in Fig 5. converted to six-phase.1 6. but delivering same amount of total load of 1320MVA. The current in a single conductor is 577A. Table 5.8.38 22.34 22.2.Chapter 5 Electromagnetic Field Gradients While moving away from the transmission line the magnetic field decreases.1 6.2 Magnetic Field of Six-phase Line with same load We have the same transmission line as above.1 Yi (m) 24.86 21.1 6. 1 2 3 4 5 6 Xi (m) 6.7. 5. Figure 5. No.38 Current I (A) 577 577 577 577 577 577 Phasor (Degrees) 0o 60o 120o 180o 240o 300o The overall magnetic field profile drawn by the program for six-phase single circuit transmission line is given in Fig 5. 63 . 3: Input Data for Six-phase line with 73% increase in load Sr.2.86 21.1 6.Chapter 5 Electromagnetic Field Gradients Figure 5.1 6.1 Yi (m) 24. 1 2 3 4 5 6 Xi (m) 6.38 Current I (A) 1000 1000 1000 1000 1000 1000 Phasor (Degrees) 0o 60o 120o 180o 240o 300o The overall magnetic field profile drawn by the program for six-phase single circuit transmission line is given in Fig 5.10.8: Magnetic field Profile of Six-phase line with same Load 5.34 21. but delivering 73% extra load than delivered by three-phase double circuit line. The input data to the computer program is given in Table 5. Table 5.3. converted to six-phase.86 24.34 22.1 6. No.1 6.38 22. 64 . The load is 2283MVA and the current in a single conductor is 1000A.3 Magnetic Field of Six-phase Line with Increased load We have the same transmission line as above.1 6. 10.Chapter 5 Electromagnetic Field Gradients Figure 5. Magnetic field around six-phase transmission line with same load is less than three-phase double circuit transmission line. 65 .2. following results are obvious: 1. the plot of magnetic field strength versus the distance along a slope of 2 is plotted in Figure 5. Figure 5-10: Plot of Magnetic field of six-phase line 5. 2.9: Magnetic Field Profile of Six-Phase with increased load While moving away from the transmission line the magnetic field decreases.4 Results and Conclusion From above plots. Magnetic field of three-phase double circuit transmission line is concentrated near the conductors and has greater strength between the conductors. 3. 4. the electric field is considered as quasi static although they vary sinusoidally with time at power frequency. Magnetic field of the six-phase line sustains a very small value. so less conductor spacing is required in six-phase conductors. and then it decays slowly. But its strength between conductors in less than that of three-phase double circuit line.3.Chapter 5 Electromagnetic Field Gradients 3. Magnetic field around six-phase line is decreases rapidly in start. For DC transmission line the electric field is purely static field in case of AC transmission line. Six-phase line magnetic field is less concentrated but sustains long as we move away from transmission line. even after 20m from the center of the line it sustains a little amount of magnetic field strength. 5.1 Introduction of Electric fields The voltage applied to the conductor of a transmission line produce electric field in the region around the conductor and of course between the conductor and ground. 5. Magnetic field strength between the conductors is less for a six-phase transmission line as compared to that of three-phase double circuit transmission line. From above statements following conclusions can be made: 1. 6. 7. six-phase transmission line have no trouble in feasibility regarding magnetic field concerns. Magnetic field strength between the conductors of six-phase line is less than that of three-phase line. but it has a benefit that compact structures can be made that require less conductor spacing. 2. So. Magnetic Field of a six-phase line with increased load is greater than magnetic field in three-phase double circuit line. even after 20m distance from the center of line.3 Analysis of transmission line conductor surface voltage gradients computations 5. Thus the frequency of variation of the field is sufficiently low to permit the consideration of the electric field 66 . but the value is less than 2A/m which is environmentally safe. Magnetic field around three-phase double circuit transmission line vanishes rapidly and disappears completely after a distance of 20m from line. horizontal. where H is the height above ground at the support point and S is the conductor sag corresponding to the mean annual temperature. conducting plane surface. proximity of towers. certain basic assumptions are involved in all existing methods for calculating the electric field in the vicinity of transmission line conductors. The results of this experiment are described by coulomb’s law and given by: ⃗ ̂ (5. The analysis of transmission line conductor surface voltage gradients requires an understanding of the basic assumption and theorems. The ground is assumed to be an infinite. finite ground conductivity etc.2 Basic Equations It has been found experimentally that the force between two stationary electric point charges Qa and Qb a) acts along the line joining the two charges.15) Where F = force. infinitely long circular cylinders parallel to each other and to the ground plane. The influence of the conductor support structures and of any objects in the vicinity of the conductors is neglected. uneven conductor and ground surface. b) is proportional to the product Qa*Qb c) is inversely proportional to the square of the distance ‘’r’’ separating the charges. let’s start with coulomb’s law for electrostatic forces because it is fundamental. Qb = charges. It is used then to derive Laplace’s equation which makes it possible to calculate the electric field strength and voltage produced by transmission lines one of the most useful tools used in these. coulombs 67 . 5. Before beginning the study of electromagnetic fields by investigating those fields that originate from stationary electric charges. The conductors are assumed to be smooth.Chapter 5 Electromagnetic Field Gradients independently of each other and calculation on the basis of static field concepts.3. The horizontal spacing between the conductors remains constant at a specified value and the height above ground of each conductor is an average value equal to H + 2/3 s. Newton ̂ = unit vector pointing in direction Qa. The calculation of the electric field produced by transmission lines is a complex problem because of the following practical aspects: conductor sag. The and the field of electric filed intensity E is defined to be the force per unit charge exerted on a test charge in the field.3 is angle between E and 68 .Chapter 5 = permittivity of the medium.17) is measured. and resultant E is simply the vector sum of the entire individual E’s by the principle of superposition.16) is If the electric field is produced by more than one charge each one produces its own filed.11: Vector addition of fields due to two charges. The total field E is the vector addition of field due to individual charges. farads/meter r = distance between charges.17) Referring to the Fig 5. The direction of E is distance from i to a test point where is defined as pointing away from a positive charge and towards a negative charge as Figure 5. It is proportional to the electric field strength E and to the distance the charge is moved parallel to the field. Thus ⃗⃗ Where shown ∑ ̂ (5. The electric potential is defined as the work required moving it per unit charge. Consider a test point charge Q that can be moved from a to b in an electric field E. Thus the electric filed intensity due to the point charge ⃗⃗⃗⃗⃗ ⃗ ⁄ (5. That is ∫ ∫ ∫ (5. meters The interaction between point charges consiered as an interaction between Electromagnetic Field Gradients and in coulomb’s law can be or vice versa. from a positive point charge.15 become (5.This gives a zero potential at r2 and Equation 5.As shown in Fig .12: Potential difference between two points a and b In case of a nonuniform field. ∫ (5.14) that the electric field E is given by the negative rate .a test charge is moved from radius r2 to radius r1.18) Figure 5.15 by placing r2 at infinity.the electric field E is radial and is inversaly propotional to squre of the distance r from the source charge.Chapter 5 Electromagnetic Field Gradients Figure 5.19) This is called the absolute potential of the point 69 due to the charge Q.The potential difference between the points is given by inseritng Equation (5.it was pointed out in Equation (5.13:Linear path in nonunform electric field The potential at r1 can be calculated from Equation 5.17).16) into (5. The Laplace operator gives (5. and D is flux density. The del operator is then defined as a vector operation. it can be 70 . This statement is abbreviated to expression E=-grad V or V.22) (5.25) Applying the divergence theorem if we replace E by in above equation the. In rectangular coordinates (5.24 can be rewritten as ∮ ∮ (5.28) This is Laplace’s Equation. Gauss’s law provides us with a powerful method for calculating the electric field intensity E of simple charge distribution.26) This is Poisson’s equation.23) According to gauss’s law the flux of E through a closed surface equals the total charge enclosed within the surface. Since Equation 5.21) In cylindrical coordinates (5. And is called either electric field strength or voltage gradient. The general problem of finding the electrical potential V corresponding to a given charge distribution amounts to finding a solution of either Laplace’s or Poisson’s equation that will satisfy the given boundary condition. (5. When Laplace’s equation is applied to a transmission line. Gauss’s law stated in integral form is written as ∫ Where ∫ (5.27) In the region of field where the charge density 𝝆 is zero (5.Chapter 5 Electromagnetic Field Gradients of charge or the negative gradient of potential reduces most rapidly.20) (5.24) charge density is is the volume enclosed by the surface S. Chapter 5 Electromagnetic Field Gradients solved either directly or numerically to give the voltage near line. The simplification used because virtually no loss of accuracy and computational difficulty is reduced greatly [20]. 5.3.3 Conductor Surface Electric field strength Considering now a n-conductor transmission line. It can be represented by n infinitely-long cylindrical conductors or radii r1 r2…. Rn, placed parallel to and at heights of h1, h2,….hn above the ground plane, as shown in figure 2.8. by using the ground plane for imaging, the problem is transformed into that of solving the electric field of pairs of parallel cylindrical conductors in finite space with equal and opposite voltage applied to them. The analysis shows that the charge distribution on each conductor can be represented by means of a line charge located at a small distance away from the center of the conductor. The distance is a direct function of H/r. for large values of H/r, as in the case of practical transmission line configuration the line charge is located very close to the center of the conductor. (⃗ ) Here V = column vector of n complex line-to-line voltages, volts Q = column vector of n complex line charges, P = n x matrix of potential coefficients, (⃗ ) Figure 5.14: Transmission line of n-conductors 71 Chapter 5 Electromagnetic Field Gradients In order to simplify the inherently complex problem of calculating the electric field 1. The ground is assumed to be infinite horizontal 2. Conductor are assumed to be equipotential Generally, a practical high voltage transmission line conductor can consist of several sub conductors. In this case, each conductor bundle is replaced by a single conductor with an equivalent capacitance the radius of the equivalent conductor for a regular bundle of n sub conductors is Where n = number of sub conductors r = sub conductor radius, meter R = bundle radius, meter The simplification will not affect the accuracy of results at or near ground level, even though accurate calculation of voltage gradients at the conductor surface cannot be made using this model. As we have discussed is the previous section the electric field strength at radius r from an infinite line. ∫ ∫ (5.29) Considering the system in fig based on the theory of images the ground plane may be replaced by an image conductor of radius r located at a distance H below the ground. The heights of conductor above ground as well as the distance between the individual conductors are very large compared to radii of conductors. Therefore the charge on each conductor is then represented by a line charge located at its center. The potential at a point on the surface of conductor in Fig is expressed as that produced by the line charge Q and its image charge –Q. 72 Chapter 5 Electromagnetic Field Gradients Figure 5.15: Electric fireld produced by source and image conductor The electric strength at any point p (Xp,Yp) near the ground plane can be determined as that produced by the line charge Q and its image –Q. From equation the electric field component E1 produced by +Q is directed along the line joining the centre of conductor and point P and has a magnitude | | (5.30) (5.31) Where (5.32) X and Y components of E1 are obtained as | | | | (5.33) (5.34) Similarly the electric field components is E2 produced by the image charge –Q and has magnitude | Where, | (5.35) 73 (X2. Let V1.…….. (5. With reference to arbitrary coordinate system the coordinate of n conductor are represented by (X1.37) (5.44) ) ( ) ) (5. (X2.Vn be voltage applied and Q1.Y1).-Q2. (Xn.41) Where effective value of voltage and w is is angular velocity.R2.-V2.-Y2).H2.….Yn).……………..V2.………Qn be line charges representing and located at the center of conductor. Now we expand the system in Fig 5.-Yn) and having potential of –V1. above the ground plane as shown. (Xn.9 to n conductor of radii R1.in case of AC line the voltage varies with time (5. The image conductor are expressed by line charges of –Q1.42) (5.-Y1)..43) .39) (5.Rn and placed parallel at heights of H1.. ( √( Where.……-Vn.-Qn. Using the principle of superposition and equation (5.40) And the magnitude and direction of | | √ are .38) Thus the resulting electric field Ep at point P is obtained by adding the X and Y components.36) (5. √( ) ( 74 ) (5.Hn. ̅̅̅ For the DC line the potential V is contents and the electric field at any point is defined by a space vector having a constants magnitude and direction as given below . applying theory of image the ground plane is replaced by image conductor located at (X1.…….…..31) we can write equation for conductor potential.Y2).Chapter 5 √ | | | | Electromagnetic Field Gradients (5.Q2.….. Since the line voltage are generally known .45 and 5.48) ) (5. conductors. the bundle equivalent radius.15 and 5. Equation (5.42) is same as Equation (⃗ ) (⃗ ) The formula for diagonal and off diagonal elements of the potential coefficient matrix P is (5.and the potential coefficient can be determined from the line geometry by using equ.16: n-conductor system Rewritten in matrix form. produced by the line charge Qi and its image are obtained ̅ (| ̅ | | | ) | (5.46) In equation 5.47) is replaced with .49) (| | | 75 .Chapter 5 Electromagnetic Field Gradients Figure 5.45.46 the line charges Q can be obtained by solving equation = (5.5.Yp) between the conductor and the ground. for bundle Following equ.45) (5.5.42 the X and Y components of the electric field strength at any point P(Xp. (x1. This function is called again and again for 501x501 points to evaluate magnetic field.17 76 .3. Tower geometry to the program is given by the six points. voltage magnitudes and the space (set of points) in which the electric field is to be evaluated. 5. y2). 5. the resultent potential and electric field strength at P are ∑ √ ∑ (| | | | (5.42..….5.50) ) (5.53 is the required equation for calculation of electric field at a pont P. Electromagnetic Field Gradients For n conductor the resultant X and Y components of the electric field strength at P are then obtained as ∑ ∑ ∑ ∑ ̅( | | | | ) | (5. The program is divided into two functions. a plot of magnetic field strength versus distance from transmission line and a complete profile of transmission line magnetic field strength. The resultant matrix is then plotted against the x and y arrays to form a magnetic field profile in the form of contours. The overall electric field profile drawn by the program for three-phase double circuit transmission line and six-phase single circuit line is given in Fig.51) ̅( | | | Finally.4 Computer Program for calculation of Electric Fields A program in MATLAB is written for the calculation of electric field. One of the functions is called by the other function gives magnetic field strength against a single point input in the form of x and y coordinates of the point.Chapter 5 Where are defined as in Equation 5. Tower geometry representing these points is given in Figure 5.53) Equation 5. The computer Program is listed in Appendix B. y1). y6). The input to the program is the geometry of the tower. (x2.52) ) (5. The output of the program is the value of electric field strengths.(x6. Figure 5.Chapter 5 Electromagnetic Field Gradients Figure 5.18: Plot of Electric Field versus Distance for Three-Phase In case of six-phase single circuit transmission line becomes equal to the .17: Electric field profiles In case of three phase double circuit the So by using v=500 in the MATLAB code the electric fields magnitudes graph is shown below. reduces by √ times and So by using 500/√ in MATLAB code the following plot appears. 77 . one for convergence of potentials and another for convergence of electric field at conductor surface. instead of using flux-tube and writing continuity current equation along it. In this method only one loop is needed for convergence of space charge density. Boundary Element Method.19: Plot of Electric Field versus Distance for Six-Phase The geometry of the tower is same as it is given in Table 5. for updating space 78 . 5. such as using new updating space charge densities along electric field lines.4 Corona One of problem associated with HVDC and HVAC transmission lines is corona power loss. and Finite Element Method. however.Chapter 5 Electromagnetic Field Gradients Figure 5. etc. Many attempts were made to solve ionized field using Charge Simulation Method (CSM). in previous method deal only with the potentials in conductor and ground plane and check the field on conductor surface later. In previous method programming calls for two loops to convergence.1 that has also been used for the calculations of magnetic fields. but some innovations. But none of them has been taken in account the effect of the diffusion coefficient as function of electric field and climate temperature and air density. this of course reduces the complexity if computation and leads to reduction of the number of iterations. The latest method for corona power loss calculation is FEM method that is used in this paper. The present method implements the potentials and electric field at conductor surface as boundary conditions. 12 (wet) to 0. 𝛿= (a) where b = atmospheric pressure (cm of mercury) T = atmospheric temperature (°C) The equation for = ( √ ) m (1+ √ ) (b) where m = conductor surface factor. the rung-kutta integration method is used to calculate charge densities along electric field lines. OF CIRCULTS= 1 NO. using flux-tubes along electric field lines [21].00 KV NO. 79 . 0. OF GROUND WIRES= 2 EARTH RESISTIVITY= 100. OF SUBCONDUCTORS PER PHASE= I TOTAL NO. varying between 0. = (f+25) √ ( kW/mile/conductor where f = system frequency (Hz) GMD = Geometric mean distance (cm) = 15th root of all fifteen combinations of distance between the conductors of a sixphase line E = Maximum surface gradient (kV/cm) 𝛿 = Relative air density given by (a) r = outside radius of conductor (cm) = Corona initiation gradient (kV/cm) given by (b) Now.1 Corona loss Calculations The corona loss in a six-phase line can be obtained using the following empirical formula.4.00 HZ BASE POWER= 100. Radius (FT) GMR (FT) A - 0.4: Line Configuration and Conductor Data CONDUCTOR DESIGNATION Horizontal Spacing (FT) Height at Tower (FT) 68.00 KV Table 5. whereas previous method. OF PHASES= 6 NO. 5. which is valid for three-phase lines also.Chapter 5 Electromagnetic Field Gradients charge densities around the conductor.00 MVA BASE VOLTAGE= 138.00 OHM-METERS FREQUENCY= 60. Mid-Span Clearance (FT) 56. Basic System Description Data: SYSTEM VOLTAGE= 138.96 (dry). 0000 GR1 -6. 2000 Temperature (°C) Surface Factor (CONSTANT) Critical Gradient (KV/CM) Corona Loss (KW/Mile) 21. 1111 21. 2000 76. 8000 0.0000 77. 3647 8. 2000 76. 0019 0. 0000 E 14 .9880 525. 0000 68. 5000 GR2 6.0000 77. 5000 Table 5. 0470 10. 1111 21. 6000 0.3502 0 0 0 0 The above results are obtained using the EPPC. indicating greater than expected corona activity.Chapter 5 11. 1111 21. 0940 18. 0143 0. 0143 0386 0. 4117 16. 5000 0. 0386 0. 9000 2. 4000 0.5: Results for corona loss Electromagnetic Field Gradients 0000 43. 0386 0. 1411 914. 0484 0.7129 243. 0484 0. 1111 21. 1111 21. 4587 24. 0484 0. 7000 0. 5. 0386 0. 0000 30. 6823 5. 1111 21. 2000 76. 0484 0.0000 B 14. 1000 0. 0000 42.4. 1111 21. 2000 76. 7764 21. the measured 1 megahertz (MHz) radio noise was higher than expected based on preconstruction calculations.0000 D 11. 0000 55.5000 C 11. 2000 76. 0386 0.5000 F 11.0000 0000 55. 0019 Barometric pressure (CM of HG) 76. 2000 76. 1000 0484 0.6815 68. 0000 67.a computer program for six-phase transmission line design [22]. 1111 21. 1111 0.8939 1. 2000 76. 0000 30. 0000 56. 0000 43. 7294 13. 80 . 3000 0. 0484 0. 0000 42. 1000 67. 2000 0. 0386 0.2 Corona Precautions for Compact Lines When the Goudey-Oakdale line was first energized at 93 kV six-phase. 2000 76.  With the bucket alongside the spacer. a piece of hardware may test successfully in the laboratory. It was not possible to determine which conductors were primarily contributing to the sound heard by ear. the electric field is greater for the same voltage. It is frequently necessary to specify EHV-type corona free hardware for use on compact 115 kV lines. instead of the corona-free variety. because it is based on voltage. The bottom two phases clearly manifested significantly greater electrical discharge of a different type than the other four phases. This method has worked well for many years. not electric field. but may have excessive corona in actual operation. fiber optic cable was initially wrapped on the bottom two phases the entire length from Goudey to Oakdale. These tests were performed from a bucket at approximately 45 feet above the ground. the noise was greater pointing at the spacer than at the tower.Chapter 5 Electromagnetic Field Gradients Corona is a function of the electric field at the surface of the conductor and hardware. Traditional laboratory tests for corona acceptability involve setting up a specimen in a laboratory and energizing at some percentage above normal operating voltage to check for corona inception. However. with the maximum at the spacers. These were suspected of contributing to the elevated radio noise levels. Thus. the ultrasonic noise was greatest off the bottom phases. it is not a complete test. Also. Measurements with these types of instruments taken in the compact section revealed the following:  The ultrasonic detector revealed a raspy sound from the bottom two phases similar to gap discharges. When conductors are more closely spaced than conventional design.  Using the ultrasonic detector from the bucket between the tower and the first in-span spacer. because the electric field stress is actually more typical of a 345 kV line than a 115 kV line I ' The conductor shoes used on the original Wshaped spacers installed in the compact section were of the standard design with screw threads and nuts protruding from the bottom of the clamps. There was also a little noise off the upper four phases which seemed to be coming from the ends of 81 . An ultrasonic sound detector and VHF radio receiver were used to verify that the spacer hardware and fiber optic cable were contributing to the elevated noise levels. At this elevation unaided audible noise could be heard in fair weather coming off the line. 5. corona loss.4. 82 . TV interference and formation of ozone due to corona will also reduce as electric field strengths are diminished. 5. sixphase line has lower electromagnetic fields and corona loss. on the basis of electric field.5 Summary In this chapter electromagnetic field gradients of a transmission line have been discussed.  245 MHz noise peaked with a directional antenna pointed at the in-span spacers from the bucket located between the tower and the first spacer. Computer programs are written in MATLAB and are used for plotting the profiles of electric and magnetic fields in both three and six-phase transmission lines. It was not possible to distinguish the relative level of noise from the different phases [23].18 clearly indicates that magnitude of electric field in case of six-phase transmission is relatively smaller in magnitude.3 Results Fig. Eventually. No noise was detected from the ends of the armor rods. radio interference.Chapter 5 Electromagnetic Field Gradients the conductor clamps of the spacers. That is. 5.17 and fig. From this it can be concluded that size of insulator required in six phase transmission towers will be less as compared to the three-phase double circuit and size of tower will also be compact as ground clearances and mid span clearances will be reduced. The results are in favor of six-phase transmission line. 5. corona loss in the transmission line is determined. Finally. After power transmission been analyzed by using the six phase system. Studies performed prove that transmission line with six phase system has several advantages as high phase transmission line. audio noise. the use of more than three phases for power transmission. the modifications required in conversion of a threephase double circuit transmission line to a six-phase lines are discussed and discussing the savings/expenses in terms of cost in all the equipment. However. has been extensively studied in the last ten years. In this chapter. for a technology to be applied. Six phase transmission line system get enhance the capability delivery as many 73% over with double circuit three-phase system. Later a cost analysis is performed in which a 500kV six-phase line is compared for relative economics with a 500 kV three-phase double circuit design. By implement small development structure on the system use. 83 .Chapter 6 System Modifications and Cost Analysis Chapter 6 System Modifications and Cost Analysis High phase order. television and radio interference and magnetic field giving good impact to the environmental. Increased power transfer over existing rights of way and reduced electrical environmental impact are two of these benefits. it must be economically as well as technically beneficial. For reduction of corona effect. it can be concluded that. it found affordable to enhance the capability overhead line space on it system advantages. A number of papers and reports have presented technical characteristics and benefits to be obtained by the use of more than three phases. Six-phase has already been shown to be an economic uprating tool for double circuit lines. Number phase increase will cause reduction gradient conductor surface. the six phase system is one an alternative to replace double three phase circuit. It can either be constructed by using two three-phase transformers or six single phase transformers.1. So. the installation of new three-phase transformers is justifiable.1 Six-Phase Conversion Transformers As discussed in chapter 3. but when electrically configured into a six phase system. 6. since it will have greater than normal impact on the physical arrangements. the existing transformers can be used in forming a transformer conversion bank.[24] Proper phasing is exercised to ensure that each three phase conductor subset of the six-phase system is connected to the 84 . 6.2 Six Phase Positioning Design of Substation modifications for six phase transmission requires careful attention to detail regarding the phasing arrangement. In vector the two sets are 180o out of phase. So in using the existing transformers in forming the six-phase conversion bank there is saving of three single phases (or a three phase) transformers. At high voltage levels usually three-single phase transformers are used to form the three-phase transformer. so in six-phase conversion.Chapter 6 System Modifications and Cost Analysis 6. then another three-phase need to be purchased.1. Since the six-phase configuration is mostly achieved through the use of two three-phase transformers using Delta-Wye and Delta-Inverted Wye configurations. the phasing arrangement of the six phase system (1-2-3-4-5-6) can be visualized as being built with subsets of two three phase systems. it will result in vector displacement of 60 degrees between adjacent phases. one set comprised of phase 1-3-5 and the second comprised of phase 2-4-6. In case a three phase transformer is installed. As the power transmission capacity is being enhanced. In practical conversion of three-phase double circuit transmission line to six-phase line requires the installation of new six-phase transformers. the most suitable way for the production of six-phase is by using three-phase to six-phase conversion transformer.1 System Modifications In this section major modifications required in the power transmission system in conversion of a three-phase double circuit transmission line to a six-phase transmission line are discussed. Further. three more transformers need to be installed. [18] However depending upon the life. the transformer required must be of higher rating to meet the enhanced power flow. the transformers of higher rating are always needed even if the method of power transfer capability enhancement is other than six-phase transmission. 1. and the types of shortcircuit faults are as many as 120 in species that is only 11 in three-phase system. In addition. Phase transposition at the transmission tower itself utilizing additional insulator strings and cross over jumpers to achieve the designated vector configuration. Seeking the complexity of protection in six-phase transmission system it must be given attention.3 Six-phase Bays Apart from the need for the positioning structures transformer and switching bays are required for six-phase at high voltages. Reference [26] accurately expressed the symmetrical arrangement electromagnetic coupling sequence and derived the fault current expression based on the analysis of various fault types of six-phase system.4 Protection The concept of protection in six-phase is entirely different from that of three-phase transmission system. it requires the installation of intelligent and sophisticated protection equipment for current differential line protection and appropriate 85 . adequate clearances are maintained during phase transposition of conductors at the secondaries of transformers since the voltage difference between adjacent phases conductors could be 1 pu or 1. foreign scholars have conducted some research on the six-phase transmission system faults. 6.1. Phase transposition from the transformers to the first transmission tower of the six phase line could have been achieved in one of two ways: 1.73 pu or 2 pu as discussed in chapter 2. the number of significant faults in six-phase is 23 whereas in three-phase are only 5. 2. Faults in six-phase transmission lines are much more complicated than that of three-phase transmission lines. These structures are similar to that of the phase positioning structures and also have a similar impact on cost. Currently. This was a critical issue for the conductor connections between the transformer take-off structure and first tower of transmission line. 6. So it requires space and structures for the said purpose at the substation with extension is the ground grid that ultimately appears in the form of increase in cost.Chapter 6 System Modifications and Cost Analysis appropriate transformer terminals at each end of the line. Further. [25] These bays provide housing for the six-phase transformers and circuit breakers respectively. Provision of phase transposition buses on top of the takeoff structure. So. So. This is again saving in terms of cost.7 Tower Structures Transmission towers are priced according to their weight. which is the major cost saving in six-phase transmission system.1. That is the power transferring is under the same structures. Three-phase double circuit transmission line to six-phase transmission line reduces the requirements of supporting structures. there is no need to reconductoring nor are the excessive insulators required. Research shows that if proper [27] positioning of all the six-phases is done on an existing three-phase double circuit transmission tower.6 Insulation Requirements We know that in six-phase transmission line. 6. there is no change in the line currents.Chapter 6 System Modifications and Cost Analysis transformer protection. [28] Transmission towers are designed to carry the load of the conductors hanged with the insulators. the phase to ground voltage increases to line to line voltage. that is the line to line voltage is reduced to phase to ground voltages. as a result the steel structures required to carry the conductor and insulators reduced. Also requires specially modified auto reclosing and synchronizing relays. Thus the system voltage is again the same as in three-phase double circuit transmission line. breaker failure protection for each line breaker.5 Transmission line Modifications In uprating three-phase double circuit transmission line to six-phase transmission line. So. that even after converting a threephase double circuit transmission line to six-phase single circuit transmission line.1. This is a costing factor in six-phase system. the insulation requirements considerably reduce due to reduction in the system voltage. tower weight is a primary parameter in the economic analysis. If the load is not increased in conversion from three-phase double circuit transmission line to six-phase transmission line. It is proved in chapter 4. 6. Calculations done in chapter 2 showed that the maximum potential that exists between any two phases in a six-phase transmission system is not more than 2 p. 6. pole disagreement protection. 73% power enhancement is achieved.u. there is no need to provide extra insulators on transmission line. due to 86 . As in three-phase to six phase conversion. Each tower was fully designed and all members properly sized.1. Thus. Further. The currents in each phase have the same magnitudes as before in three-phase double circuit transmission line conductors. there is no need of re-conductoring. and metering. reduced electromagnetic field gradients results in compact structures allow a lot of saving in constructing a new six-phase transmission line. Further. the re-conductoring of existing line with a conductor having greater thermal capacity. The existing substation was assumed to be a 500 kV breaker and a half arrangement with 1200 ampere rated equipment. the cost of ROW varies widely for different locations and areas. and can result in a significant cost advantage [29]. in constructing a new transmission six-phase line in comparison to three-phase double circuit line has a lot of saving in terms of capital required. The system was assumed to require uprating to carry an additional 900 MVA. For this Economic Analysis. One option to obtain a rating increase would be to reconductor the line with 795 kcmil ACSR conductor by bundling with the existing conductor. it would not be correct to assume that all utilities would use the minimum. cost differences for the candidate lines studied are presented only as a single illustration. it is assumed that there is a double circuit 500 kV line between two substations which requires an increase in power flow capability. and are not included in the general case.8 Right of Ways The required width of Right of Way is based on electrical system parameters. 6. Which is to be uprated to carry some extra load.e. So. For the assumed 87 .2 Cost Analysis In this section we introduce a transmission line carrying a specific amount of load. the Right of Way span required for six-phase power transmission line is not greater than that of three-phase transmission line. with combined capability for the two circuits of 1400 MVA. For these reasons. Also. The ROW requirements of EHV high phase order lines are less than those of three-phase double circuit transmission lines. As discussed in previous chapters that electric and magnetic field gradients of a threephase to six-phase converted transmission line are within the limits governed by the health authorities. 6.Chapter 6 System Modifications and Cost Analysis lesser electric field in six-phase power transmission. the less spacing requirement between the six-phase conductors results in the smaller arms of transmission tower.1. The existing line was assumed to be constructed with 795 kcmil ACSR (Drake) conductors. While there is a minimum required width for any transmission line. making it difficult to assign a meaningful dollar-per-acre ROW cost. So. the options considered are the sixphase conversion and other cheapest possible i. Re-conductoring increased the line's thermal capacity to 2800 MVA.1: Cost for the Equipment to be installed in Six-Phase line Sr. Costs of for different equipment are listed below. No.Chapter 6 System Modifications and Cost Analysis line and substations. 1 2 3 Equipment Line Feeder Bay (4) Transformer Bay (4) Single Phase Transformers (6) (400MVA Each) Cost (Per Unit) $ 382075 $ 280550 $ 541600 Total Cost $ 1528300 $ 1122200 $ 3249600 4 Line ( km) $ 73500/km $ 73500 $ 5900100 + 73500 Total Cost This is a typical up gradation cost of three-phase double circuit transmission line to six-phase transmission line. reinforcement of tangent structures. in this cost analysis. replacement of dead end and angle structures. So. by reconductoring the line. which may be a consideration for longer lines. 88 . However. and new line hardware. In which three single phase transformers are assumed to form three-phase transformer formerly. It is quite easily observable from the two tables that the terminal equipment in sixphase line is more costly in than a three-phase line. These costs are obtained from the reference [30]. this option do require substation modifications. the six already used transformers at both ends are used. So it appears to be more feasible for long length transmission lines. there is a saving in the transmission line equipment. Now for the sake of comparison we also take the uprating of an existing three-phase double circuit transmission line to enhance the power transfer capability. The six-phase conversion would result in higher surge impedance loading. Table 6. These costs were available in South African Rand and are converted to US dollars. using two three-phase (six single-phase) transformers for phase conversion at each end. The other option was to convert the line to operate at 500 kV six-phase. and the six-phase conversion increased the thermal capacity to 2420 MVA. Both options therefore gave similar thermal ratings. only six new transformers are installed three at each end. The cost of the equipment to be installed is given in the following table. the length of line where the cost of six-phase line is equal to that of three-phase double circuit transmission line. Figure 6.Chapter 6 Sr. No. the total cost equation for six-phase and threephase double circuit transmission line is given. Figure shows that breakeven distance occurs at 9km. for this particular line the six-phase configuration of transmission line capability is enhanced only if the line has a length greater than 9km. 89 .1: Plot of Total Line Costs for Six-phase and three-phase double circuit lines.e. i. $ 520150 $ 376584 $ 1992700 $ 1040300 $ 735168 $ 3985400 $ 90780/km $ 90780 $ 5760868 + 90780 Total Cost At the end of the two tables above.2: Cost of the equipment for uprating of three-phase double circuit line. This is the minimum length for which the six-phase line is beneficial. These equations are equated to give the breakeven distance. 1 2 3 Line Feeder Bay (2) Transformer Bay (2) Three.Phase Transformers (2) (400MVA Each) 4 Line ( km) Equipment System Modifications and Cost Analysis Cost (Per Unit) Total Cost Table 6. So. Saving and costs in substation equipment.Chapter 6 System Modifications and Cost Analysis 6.3 Summary In this chapter various system modifications are discussed that are needed in uprating an existing three-phase double circuit transmission line. 90 . protection equipment and lines are discussed. At the end a cost comparison of six-phase transmission line made with a three-phase transmission line and it was found that a 500kV transmission line has a benefit is uprating to six-phase line in terms of cost if the line length is more than 6 km. 1 Results and Conclusions Following results and conclusions can be made from the calculations. 1. Where analyses have been performed for the power enhancement capability. 7. In three phase to six-phase conversion. Moreover. After establishing basics. where the complexity in different voltages in six-phase system is discussed in detail. A comparison of three-phase double circuit transmission line with six-phase transmission line is done. The voltage drop along the length of a six-phase transmission line is also discussed as a comparison with three-phase double circuit transmission line. line to line voltage can be made equal to phase to ground voltage or alternatively phase to ground voltage 91 . Later on. methods have been established for the production of six-phase power and analysis has been performed on the three-phase to six-phase conversion transformers. In this chapter the results of all the analysis performed are summarized and discussed.Chapter 7 Conclusions and Future Recommendations Chapter 7 Conclusions and Future Recommendations This thesis provides a base for the six-phase transmission system and it explains some basic rules about the six phase power. The conclusions are made and limitations of this study are discussed and finally recommendations are made for further study in this field. six-phase transmission line is modeled in MATLAB Simulink and SimPowerSystems. Electric and Magnetic Field Gradients in a transmission line are discussed and analyzed. analyses and simulation performed in this project. checked and verified the voltage and phasor relationships developed earlier. smaller line structures. This is a drawback of six-phase power transmission line. whereas in ∆-Y & ∆-Inverted Y. 4. 3. between adjacent phases. 8. Four different types of voltages exist in Six-phase system those are phase-to-ground voltage.Chapter 7 Conclusions and Future Recommendations This voltage level transform can be increased to line to line voltage. better stability margins. magnetic fields are less than three-phase double circuit transmission line. Out of five common transformer configurations. depends upon conversion transformers. Electric field plots of six-phase transmission line are also less than those of three-phase double circuit transmission line. increased utilization of right of way. between phases separated by one intermediate phase and between opposite phases. Line Loadibility and stability can be computed by using the same techniques as used for threephase transmission line. 5. Six-phase transmission have several advantages over the three-phase transmission that include higher power transfer capability. 6. Diametrical and Double Y configuration phase to neutral voltage is increased to line to line voltage. Magnetic field plots of six-phase line show that under the same loads of as of a three-phase double circuit transmission line. Six-phase transmission line parameters such as SIL. 7. Voltage Drops across the six-phase transmission line are greater than threephase transmission line and has poor voltage regulation and poor voltage stability with increase in distance and power flow respectively. that gives a room for the compaction of transmission structures and also provide cost saving for insulators. Double Delta and Y-Y & Y-Inverted Y configuration decrease line to line voltage to make equal to phase to neutral voltage. 92 . But with 73% power enhancement magnetic fields are a bit increased but are far less than the limits set by the Environment and Health Authorities. lower corona and field effects and better lighting performance. Simulation results show that power transfer capability is enhanced in sixphase transmission in comparison to three-phase by 73% with the same line current and same line-to line voltage. increased power density. 2. less insulation requirements. PSSE etc. A more detailed 93 .2 Project Limitations and Future Recommendations Following are the project limitations and future recommendations for six phase power transmission system: 1. The accuracy of MATLAB is limited. the simulation of six-phase power transmission line is performed on MATLAB Simulink and SimPowerSystems.Chapter 7 Conclusions and Future Recommendations 9. Seeking these benefits. TV interference etc. For the same load. In this project transformer connections and their primary and secondary voltage levels are analyzed for six-phase conversion banks. For the calculations of magnetic and electric fields. 11. All these results have been verified with the help of simulations performed on Simulink® and MATLAB™ programs. So 73% more power can be transferred by the existing transmission lines without any modifications in transmission lines but the modifications are made at sending and receiving end of transmission line in the form of conversion and inversion transformer. 3. electric and magnetic fields constraints due to health hazards and provides a cost effective solution of upgrading existing transmission lines to six-phase lines. Professional power software should be used in future for more accurate results like PSCAD. 10. only modifications required are to be made at terminals. So it is concluded that six-phase transmission is a solution to the limitations offered in three-phase power transfer capability enhancement due to unavailability of right of ways. corona loss. are reduced due to which clearances required from the tower or in other sense size of tower is reduced to much extent. radio interference. Corona Loss in Six-phase is also less due to low electric field profiles. 2. a three-phase double circuit transmission line can be converted to six-phase line on the existing transmission structures. In future research can be performed on the professional tools that give three dimensional plots of field gradients with more accurate results such as EMF WORKSTATION. In this project. Cost comparison of power capability enhancement of three-phase double circuit transmission line by reconductoring to that converting to six-phase show that six-phase is more economical if line length is more than 9 kilometers. 7. dedicated tools are developed in MATLAB by our own that provide the two dimensional profile of the fields. NTDC should consider six-phase transmission lines for future transmission line construction. 4.g.Chapter 7 Conclusions and Future Recommendations studied is needed for six-phase transformers and its characteristic under sixphase operation. Research should be done at practical level by the Transmission companies by implementing hardware on a long length of transmission line e. 94 . A. S. 413-419. H. F. C.. 150. Mohsen Akbari “Voltage Stability Analysis in Conversion of Double Three phase to Six phase Transmission” Kuala Lumpur. “Composite System Reliability Evaluation Incorporating a Six-Phase Transmission Line”. 2003. Shareef. Faried. D. L. IEEE Transaction on Power Delivery. Pandya. National Conference on recent trends in power system at S. and Firuzabad. “Method of Symmetrical Coordinates Applied to the Solution of Poly-phase Networks”. October 1998. M. Proceedings of the 2002 IEEE Canadian Conference on Electrical & Computer Engineering. [6] [7] Clark.B Kelkar. Vol. [3] M.E. “Fault analysis on double three-phase to six-phase converted transmission line”. M. Institute of Science and Technology. Golkar. IEEE power engg. 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A.C.T. Brown. November 2004. x2=-6.1.1. y6=24. x1=-6. power((y5-yp).2)+ sqrt(power((xp-x2). -a*I. power((y3-yp).2)+ sqrt(power((xp-x5). y3=21. ). 2) 2) 2) 2) 2) 2) ). Ii4=0*I. y4=21. ).2)+ sqrt(power((xp-x6). -a*I. Ii2=-b*I. y2=22.1. Ii5=b*I. ). power((y6-yp).34. power((y2-yp). x6=6. ).1.1 Function for magnetic fields xp=0. y1=24. Ii3=-b*I. x4=6.1.2)+ = = = = = = I*1. Ii6=b*I. x3=-6.38.86. -1*I. yp=0. b=sqrt(3)/2. x5=6. 98 . power((y1-yp). I=1000. Ii1=0*I.Appendix A MATLAB Function for magnetic field calculation Appendix A MATLAB function for magnetic field calculations A.2)+ sqrt(power((xp-x4). I*a. y5=22.2)+ sqrt(power((xp-x3).86.1.34. a=1/2.38. ). R1= R2= R3= R4= R5= R6= Ir1 Ir2 Ir3 Ir4 Ir5 Ir6 sqrt(power((xp-x1). a*I. power((y4-yp). Hpxi1 Hpxi2 Hpxi3 Hpxi4 Hpxi5 Hpxi6 = = = = = = Ii1*(y1-yp)/(2*pi*power(R1. 2)). Hpyi1 Hpyi2 Hpyi3 Hpyi4 Hpyi5 Hpyi6 = = = = = = Ii1*(xp-x1)/(2*pi*power(R1. 2)). 2)). Ir6*(y6-yp)/(2*pi*power(R6. 2)). 2)). Ir2*(y2-yp)/(2*pi*power(R2.e).1:50. Ir2*(xp-x2)/(2*pi*power(R2.e)= magnet14 (x(1. 99 . 2)). 2)).2 Function for Plotting Profiles x=-25:0. 2)). 1. H=zeros(501. 2)). 2)). Ir6*(xp-x6)/(2*pi*power(R6. Ir5*(xp-x5)/(2*pi*power(R5.1:25. 2)).501). Ii2*(y2-yp)/(2*pi*power(R2.1:50 x=(y-22. Ir4*(y4-yp)/(2*pi*power(R4. Hpx = Hpxr + j*Hpxi Hpy = Hpxr + j*Hpyi Hx=[Hpx Hpy] A. Function for Plotting Characteristics y=0:0. 2)). Ii5*(xp-x5)/(2*pi*power(R5. 2)). y(1. Ir5*(y5-yp)/(2*pi*power(R5. 2)). 2)). 2)).Appendix A Hpxr1 Hpxr2 Hpxr3 Hpxr4 Hpxr5 Hpxr6 = = = = = = MATLAB Function for magnetic field calculation Ir1*(y1-yp)/(2*pi*power(R1. Ir3*(xp-x3)/(2*pi*power(R3. Hpyr = Hpyr1 + Hpyr2 + Hpyr3 + Hpyr4 + Hpyr5 + Hpyr6.d)). Hpxr = Hpxr1 + Hpxr2 + Hpxr3 + Hpxr4 + Hpxr5 + Hpxr6 . Ii6*(y6-yp)/(2*pi*power(R6. 2)).y. Ii2*(xp-x2)/(2*pi*power(R2. Ii4*(y4-yp)/(2*pi*power(R4. Ii3*(y3-yp)/(2*pi*power(R3. 2)). Ii5*(y5-yp)/(2*pi*power(R5.H. y=0:0.2000). Hpyr1 Hpyr2 Hpyr3 Hpyr4 Hpyr5 Hpyr6 = = = = = = Ir1*(xp-x1)/(2*pi*power(R1. 2)).86*(ones(size(y))))/(-2). Hpyi = Hpyi1 + Hpyi2 + Hpyi3 + Hpyi4 + Hpyi5 + Hpyi6. 2)). Ii6*(xp-x6)/(2*pi*power(R6. 2)). 2)). end end H figure contour(x. Ir3*(y3-yp)/(2*pi*power(R3. 2)). 2)). Ii4*(xp-x4)/(2*pi*power(R4. for d=1:501 for e=1:501 H(d. Ir4*(xp-x4)/(2*pi*power(R4. Ii3*(xp-x3)/(2*pi*power(R3. Hpxi = Hpxi1 + Hpxi2 + Hpxi3 + Hpxi4 + Hpxi5 + Hpxi6 . R(1.1) = abs(magnet14 (x(1.2)+ power((22.d).1) for d=1:501 H(d.H). y(1. 100 .d)=sqrt(power(x(1. end H plot(R.Appendix A MATLAB Function for magnetic field calculation R=zeros(size(x)) H=zeros(501. 2) ).d))).d).d)).86-y(1. 22.34]. L=zeros(6.1))^2 ). l(d. else P(d.1.8e-12))*(log(L(d. 21.e)=(1/(2*pi*8.1)-xp)^2+ (y(d.1)-y(e. 24. for d=1 : 6 for e= 1 : 6 L(d.1))^2+ (y(d. v*exp(1i*1*pi/3).86.38. lip=Lip.e)= sqrt((x(d. -6.e)/l(d.e)= sqrt((x(d.Appendix B MATLAB function for electric field calculations Appendix B MATLAB function for Electric field calculations B.1. if d ~= e P(d. v*exp(1i*3*pi/3). 21. v=500/1. ri=0.1)+y(e.86. Ep=0.d)=(1/(2*pi*8.1. 101 . Lip=zeros(6. l=L. for d=1:6 Lip(d.73. x=[-6.1)/ri)).1)-x(e.025. 6.1.1)-x(e.1].1))^2 ).8e-12))*(log(2*y(d.1. v*exp(1i*4*pi/3). y=[24. end end end V = [v*exp(1i*0*pi/3). P=L.6).1)= sqrt((x(d. v*exp(1i*2*pi/3). yp ).38. 6.1)+yp)^2 ). Q=inv(P)*V.34. -6. v*exp(1i*5*pi/3)].1))^2+ (y(d.e))). 22.1).1 Function for six-phase function [ Ep ] = elect( xp. 6. 1)= sqrt((x(d.1))^2 ).6). End B.1].1))^2 ).Appendix B MATLAB function for electric field calculations lip(d. l=L. end Ep=abs(Ep).1)= sqrt((x(d.1.1)+y(e.38.1)))^2) (Lip(d. for d=1 : 6 for e= 1 : 6 L(d. 6. 6. v*exp(1i*3*pi/3).2 Function for three-phase function [ Ep ] = elect( xp.e)= sqrt((x(d. Q=inv(P)*V.1)/ri)). -6. Ep = Ep + (-1/(2*pi*8.1.1)-x(e. lip(d.025. 22.1)*((lip(d.1).86. P=L. 21. Lip=zeros(6. -6. y=[24.d)=(1/(2*pi*8. lip=Lip. 22. yp ).1)))^2) (Lip(d.1)-xp)^2+ (y(d.e)=(1/(2*pi*8. end Ep=abs(Ep).1)-y(e.1)-yp)^2 ). 21. v=500. v*exp(1i*1*pi/3). l(d.1)-xp)^2+ (y(d. else P(d.1)-yp)^2 ). for d=1:6 Lip(d. 24.8e12))*Q(d. v*exp(1i*5*pi/3)].1. Ep = Ep + (-1/(2*pi*8.1))^2+ (y(d. x=[-6.8e-12))*(log(2*y(d.1))/((abs(lip(d.1))^2+ (y(d. v*exp(1i*4*pi/3).1)-xp)^2+ (y(d.1))/((abs(lip(d.1)/((abs(Lip(d. end end end V = [v*exp(1i*0*pi/3).1)))^2))). if d ~= e P(d.e)= sqrt((x(d. end 102 .1.1)-x(e.1.34]. Ep=0.e)/l(d. v*exp(1i*2*pi/3).86.1)= sqrt((x(d.8e12))*Q(d. ri=0. L=zeros(6.e))).1)+yp)^2 ).1)*((lip(d.8e-12))*(log(L(d.34.38.1)/((abs(Lip(d.1)))^2))). 6. R(1.H.H). for d=1:501 H(d. y(1.Appendix B MATLAB function for electric field calculations 1. Function for Plotting the Field Strengths y=0:0.1) = abs(elect (0. Function for Plotting the Profiles x=-25:0.1:50.d)). end end figure contour(x.e)= elect(x(1. y=0:0.y.e).1:25.1:50.501).2)+ power((22.d)=sqrt(power(x(1. R=zeros(size(x)). end figure plot(R. H=zeros(501. y(1.1).d))).d)).86-y(1. 103 . for d=1:501 for e=1:501 H(d. H=zeros(501. 2.2000). 2) ). x=(y-22.d).86*(ones(size(y))))/(-2).